Thanks for suggestions, everyone!
Avenger, are you still experiencing the same issue? I, too, was noticing that the searches seemed to work pretty well for the first 2 or 3 searches (that is, it would link me to whichever book I clicked on) but then would just stop linking. Also, it would not let me "refine" the search, that is, touching "refine" did nothing, just stayed on the same search results page.
I haven't noticed the problem too much in the last week or so, and I'm not sure if the difference is something Amazon did with their Kindle store on the Fire or something Hubby did. He went to the drop down screen, then Settings, then Applications, then Installed Applications. There you see a list of Apps. I think he then went to "Amazon Kindle" and clicked Force Stop and Clear Cache. Actually, to be honest, I think he did this with one or two other apps too.
Being non-tech-savvy, I'm not sure what this does, but since then, my browsing has worked pretty well. Coincidence?
There were a couple of times when I had to tap more than once, but other than that, no problems (keeping fingers crossed!) I got the idea from a problem I had with the original Kindle Fire not recognizing my books in the Cloud, and the solution I got from KB.