There is so much variability among shafts that saying “an extra stiff would help” is hard to really say. There are extra stiffs that are performing like stiff shafts and vice versa (as dave and I have encountered).
Bottom line is unless you’re swinging somewhat consistently, it’s hard to get a good fitting. Practice a bit with the same grip (not changing to try to correct something) and then get a lesson based on what feels “comfortable.” Once you’ve processed that lesson and feel good, then consider a fitting.
Only adjustment to that would be if when you get a lesson they say the fit of your current driver is a big problem, have them let you try something so you can take a lesson with a reasonable club so you can get a proper baseline.