Can long term alcohol abuse damage your motor skills and balance? And can you recover with abstinence? 3 months sober still have symptoms liver's fine

Could be a sign. Of B12 or folate (folic acid) deficiency. Talk to your doctor about testing a B12 and folate (folic acid) levels.
Maybe. Long term alcohol abuse can deplete the levels of some B vitamins (esp. thiamine) that can affect motor performance. I would strongly suggest that you get bloodwork to determine if this is the cause. Alcohol can have other effects, as well. Most of the effects return to normal with continued sobriety At your age, that is much more likely, thank goodness. Staying sober is worth it. Good job!