In the initial launch was there water
"churning" behind the prop?
I've been using a 5hp motor on my 2500kg boat as the diesel
is out of commission. It is marginal and wont successfully push the boat against all winds and seas.
But 7.5 on your boat should be able to push against a 20kt wind - albeit probably slowly.
20kts is probably at the upper limit for your set up and experience level. You should think about your launching technique. I usually have a dock "walker' who stays on the dock as I am backing out. That person steps up to the bow when the bow clears the dock and sometimes (usually if I can manage it) after the boat is shifted in to forward.
I have had engine flame outs (hence the dead diesel) after back out. And then the drama starts as we fend off other boats and try to get back to the dock.
Then I started berthing stern-to and the dock walker walks the stern to the end of the finger before stepping on board. Solo I have used a loop on the end finger cleat that I "flip" off as the stern passes the end of the finger.
Bottom line is to try not to "cast off" until you know the boat is under its own motive power.