you say you’re cross needs improvement, which you should work on first. I suggest you learn cross+1 right around the sub-10 barrier but the main thing you wanna focus on at this stage is lookahead. Turn slower. You will be able to increase your TPS as you get faster so for now, turn noticeably slower than usual (don’t turn like a turtle though). Don’t forget to look around while you are turning slower, which means if you are doing a pair, don’t look at what’s happening to it as you are inserting it and look at other parts of the cube. For example, If you are solving a pair into the back left, look for the next “obvious” pair to do. I suggest doing what i call, “blind steps” as a drill. Basically, you plan cross/cross+1, then close your eyes and solve it. Plan your next 2-3 F2L pairs, then close your eyes and solve the planned F2L pairs.
An optional thing you could do is to just not learn cross+1. The way this works is you start looking ahead during cross, which still gives a pauseless cross-F2L transition. I used this to break the sub-10 barrier. However, i do not recommend this as it is more reliable to just plan cross+1.
You can start increasing your F2L TPS when you have gotten comfortable with a slower turn speed than usual (unless you can lookahead well already)