I didn't have any of the other accessibility apps installed, but I figured out my problem.
I had turned off TalkBack in the application manager but the browser still wouldn't shut up. I went back in there and turned it back on, then went to the accessibility settings and turned the TalkBack settings on for a few minutes. When I turned the setting back off it stopped talking. This may have also been the cause of a problem I was having with one-handed mode not working. I have no idea why... but the next time I tried the one-handed swipe after doing this, it actually worked.