yes, some people are verrrry sensitive to salicylate meds and dizziness can be a symptom.
You should also try rectal meds...maybe that will be less of side effects, or you can try the steroid topical rectal meds as well.
Since your UC is nonspecific, you're in limbo. Did the docs test you for microscopic colitis?
Your dizziness could also be caused by low iron, low b12, low vitamin D, thyroid problems, low blood sugar, etc.....
You could also have a vestibular virus or a condition called BPPV (benign paroxysmal positional vertigo)...have you seen an ENT?
It's definitely best to pursue other avenues as well....but yes, it could also be the asacol, but rather than saying it's only the asacol, try to dig a bit deeper.
Keep us posted as to how you're doing.
quincy