Annie

Having Fuji, Olympus and Sony, I would advise you to try an Olympus body: its IBIS technology is fantastic and would allow you to put that "extra-shake" behind your concerns. Or, if you really want to go Fuji, I guess you would be very happy with a 2nd-hand X-H1 (I had one and replaced it with an X-T3): it is heavier and the IBIS is also great. 
But I'd check the Olympus before anything...

I agree that your shutter speeds should have eliminated any blur from camera shake at those focal lengths.  However, it looks like your auto focus just missed a bit, and you probably should have stopped down to f/8 for both deeper depth of field and I suspect that your lens is probably sharper at f/8 as well.  Finally, I wouldn't use AF-S for a moving subject unless it is only lateral movement that wouldn't change the distance between you and the subject.  

These images could not be sharp for a number of reasons. There is an easy way to check. Find a room with plenty of light, no wind, and put camera on tripod. Get one of those charts with lots of lines and colours - like a projector focussing chart - and take pictures (using a cable release or self timer) of it at various distances and apertures. Then you will know.

Not sure what went wrong for you. I have an X-T3 and do not have any real exposure problems. The XT-30 is smaller and lighter but the X-t3 is, as you say, weather resistent and has a much better viewfinder. I would say plump for the X-T3 if you can get one within your budget.

I’ve had Sony, Panasonic and Canon and prefer Fuji. With my past Canon 5D mark III my wife refused to shoot with it because of the complexities. 
Fuji isn’t as popular (or should I say: as widely discussed) as some other manufacturers. DXOMARK Still doesn’t have any information on the quality of Fuji lenses and if you study their results many manufacturers seem to have a lot of good and slightly better then good lenses and in reading lensrentals great blog, they’ve done articles on how Lens quality can differ on the same lens, copy to copy, so a serious shooter might have to experience multiple copies of the same lens to find one that was manufactured correctly.
with that, I personally think Fuji manufacturers a more quality lens then most manufacturers. My wife likes the X-T3 so much that we both have one.
 I love the image the camera produces. 
I love that I can customize the menus so much but I do find diving deep into the menus is confusing and frustrating because there are so many options and preferences in the deeper sub menus that I forget how to get to.
I find the LCD finger flick menu more of an issue then a helper. And some items marked for a quick access button don’t work and Fuji doesn’t give you a warning.
The 18-55mm Fuji kit lens you mentioned surpasses most everyone’s kit lens. 
I’ve found fuji’s 80mm 2.8 macro to be an awesomely crisp image lens.
 I rented the 8-16mm f2.8 for a week long trip to Arizona’s Grand Canyon and that’s all I shot with. It is drop dead the most excellent lens I’ve ever used.. now I’m just waiting for a sale before I buy it.
Hope that helps some... 
 

The firmware update may be coincidence. I had this experience with an x-t2. it was caused by a faulty SD card. Swopping them around will not work. The bad one will cause this fault in either slot. Replace both in the first instance and see what happens. I do hope that it is that simple for you too.

I am in the middle of  a photo safari with a camera group. Carrying 5 Fuji lenses and other gear.Was trying to upgrade  body software to v3.0 but software seemed not to be responding according to camera. After about 15 mins of no activity – tried to go back and start installation again. Could not do so as there was no indication as to how to do this. Eventually turned off camera, changed battery. Now camera will not show anything at all. No  image/message whatsoever on screen or viewfinder. Completely dead.
Used fully charged batteries with new firmware on a freshly formatted SD card.
Is there any way to wake up/restart the camera? This is a disaster as far as my trip is concerned but I still have a few days left if the camera can be restored to any kind of working order.

I had this issue the first time I rented the XT-3. The cause for me was that I had placed a SD card formatted on a Sony camera into the XT3. Once I removed the SD card, I was able to turn the camera on. 

This may or may not be the same issue that you faced.

I have been using a X-T30 for a while now, I really like it. If I had to replace it today, with a new body, I would get it again. Being tempted by the new X-T4 and its additional features would be there in my mind yes, but I manage to get enough foul weather photography fun by using basic protection for my camera. I would think again about that, if I had to routinely get photos in typhoon settings .
The only recurring issue I have had is making sure I do not accidentally change the EC dial setting, it does not lock.
I had a one time issue with an adaptor that allowed me to use a different company's long lens on this body. I had to stop, pull the battery, put the battery back in and then everything worked as it should. I never had that issue again in outings with that setup. Also, the adaptor has since had a firmware update and so I hope that does not glitch again.
Glitches ruin the fun, but I have had so very few using the X-T30, that it is easy for me to stop and ignore the problem of N=1 bias* and recommend it.
 
* N=1 bias means you do not have enough samples to make significant conclusions, your sample may be exceptionally good or bad, thus generalizing from that is bad juju.
 

Thank you, Greybeard, Jerry, and Bergat!
This all makes sense to me. I thought it was likely a settings issue. Demo'ing at a store when I haven't learned the camera yet may be less helpful than helpful I have been learning due to not knowing the camera settings and how to take advantage of the features and functions of the camera to maximize image quality.
It's good to know there is such a helpful Fuji community should I buy the XT3 (if price goes down after the XT4 goes on sale) or the XT30 (but I think I really want the weather sealing!)
Thank you!
All the best to you!
Annie
 
 

Not necessarily, the Aperture Priority setting is more concerned with only setting the shutter speed, the ISO can be set to not change or to automagically change in the camera’s set up menu. The histogram will give you a much better idea about the exposure than the exposure meter will because of the reasons Greybeard is discussing.
p.s.if you set this camera into fully manual mode, it will give you some fantastic results! Just make sure you do not accidentally move the EC dial (it does not lock!).
edit: a while back, Fujifilm issued a firmware update for the X-T3; part of the update concerned auto focus issues. It is possible the demo camera you were given to test has not had that update installed.

Its a bit difficult to tell - the focus metadata has been removed from the images when they were resized for posting. It does say that metering mode was set to average as opposed to the more intelligent multi-segment method so the sky in the second image would have skewed the exposure.

The ISO is set to 160 in these images, so that along with f11 and 1/200 is going to make for dark images. Probably for a scene this overcast, you might try starting with ISO of 400 or so for the f11 and 1/200 settings. Also, try having the histogram show in the screen, that can help you decide on exposure settings.
The camera's sensor's focus sensitivity is tied into the ISO, F number and shutter speed, (it is listed as a EV number) so in the first try of the second photo, you MAY have picked a very dark spot to try and get focus lock, but that changed during your second attempt. Without being there at that time, speculation is ..., well just that.
HTH