I understand your frustration with people walking backwards. It's pretty strong in general but there are ways to play against it. I personally had this issue because my character (Marisa) has no lows that grant a combo other than cr.lk which is never in range to hit if they're actively walking away.
So here is what I would advise:
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Actually dash forward. For a long time I basically forgot that dashing forward is a real option. Obviously you can't spam it but it should move you forward faster than them walking backward.
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Learn to value them giving up screen space. This one is a change in perspective. Them just walking backward and you walking forward should be considered a win all on it's own. They will eventually walk themselves into a corner which means better pressure and damage for you. This of course doesn't work if you let them escape the corner once there, so you need to have your antiairs sharp and ready to go. Additionally, you need to think differently about your spacing, stand somewhere that's going to make it the hardest for them to escape. Lastly for this point you need to remember that it's not a race. You have TONS of time on the clock so just be patient and don't rush things and overextend.
As for drive rush checking, if you're getting blown up by fireballs, my guess is that you're checking with too big of a button. Even if it only get a jab out of it they still spent their bar and you're likely in a decent pressure situation afterwards.