Sounds like you are doing the right things
Maybe get an EVA foam roller and / or a tennis ball and / or a lacrosse ball (if you don't already have them) for some soft tissue work. Finding knots and getting them to relax by holding sustained pressure on them gives me fairly instant mobility, anyway. Especially leaning against a ball against the wall to release round the back of my shoulder capsule and my lats (helps a lot with holding the bar in the overhead position). Stretching out the thoracic spine (lying back over a roller) helps a lot, too.
Another thing that can be important is activation. If you are like most people then you have short hip flexors and inactive glutes. I've started doing hip thrusts / bridges and x band walks to activate my glutes . That helps me drive my hips through to stand up a snatch while having the shoulder mobility to keep the bar in a balanced position on the stand up.
I spent quite some time doing 5x5 overhead squats with just the bar. Trying to get the hang of driving my hips through properly to stand the weight up. Mostly it was a stretching / mobility / recruitment exercise for me, I think. I think it is worth spending time on this and not worrying so much about the weight. Once you are activating the right muscles in the right sequence then you are in the position to strengthen the movement. I'm not terribly sure about the utility of attempting to strengthen a sub-optimal movement pattern... Especially for overhead squats since hardly anybody is limited by standing up (rather than catching in a balanced position) the snatch. You certainly want to get as strong as you can on front squats to help your cleans, though. But still... I'm considering backing off my front squat weights for a time in order to grease the groove on better hip drive... Maybe... Mixing things up can be good.