Have you tried to clear the app's cache and/or force stop it? Also uninstalling, restarting your phone and installing it again can correct bugs. If that doesn't work, then read on.
I don't use the Google messaging app. I tend not to use any of the standard apps because I've found others that I like better. There are lots of messaging apps to choose from. I am currently using a fairly simple texting app called Textra. I was using a much larger, more involved texting app but it developed all sorts of glitches and support wasn't good. Some of the glitches were similar to what you're reporting and when I used a different app, they were gone. Textra is relatively new and has really good support. I would try any other texting app and see if you have the same problem. My gut says it's the Google app & not your phone but the only way to know is to try a different app. You can actually text to yourself and see what happens if you don't have someone to test with.