AF-A and AUTO AF Area select should be fine. If not, try AF-C. It will track the subject it's locked to. If its not locking where you want, you'll have to use AF-S and Single AF Area. That requires predicting where the subject will be, and if it's going to move you'll need to set 3D to track it.
maybe some of the blur in my photos is actually better explained by camera shake (shooting at 200 mm on a 1.5x crop sensor)
This depends on which lens you are using. If it's a Nikon with VR there should be no shake. If it's not focusing fast enough, a better lens may be needed. I'd recommend the Nikon 70-300mm VR AFP f/4.5-6.3 DX E
Of course, no lens or camera can help if your problem is technique. The D7500 is a fantastic DSLR and with a suitable lens there's no reason you can't get the results you're after. Technique is crucial in that with all else accounted for, poor technique will reduce your chances.