You can help solidify the cabinet with bracing. I haven't had to do it, but looking at those...
I'd put 2 on either side of the mid like so -
View attachment 1605622
And another two in the middle of the sides 1/3 of the way up from the bottom and 1/3 of the way down from the top. Cut them so they fit tight then glue and screw. 1" dowel or 1"x1" square would be my choice.
Have you swapped sides to see if the problem follows the speaker and isn't a problem with your receiver?
Different drivers can be easy, or difficult. If you hit on the correct one right away no problems, but if the sensitivity is too low or too high you can end up with messed up sound. You might be better seeing if you can find one for cheap on ebay. Have you looked at the mid outside the cab? Sometimes there's a liner, foam, or even cups to isolate the mid from the interior cabinet vibrations. If that has come off that can cause distortion with higher volume. It's possible that the speaker has sagged, so you can also flip the speaker 180 degrees to see if that helps.