I think ‘exceedingly’ is used more in the ‘positive’ sense. For example:- She sang exceedingly well. = She sang better than expected.

And ‘excessively’ is used more in the ‘negative’ sense. For example:- He ate excessively. = He ate far too much.

But the verb ‘exceed’ can be used in either positive or negative sense.- She exceeded the standard. = She was excellent.- He exceeded the speed limit. = He drove too fast.

NB. ‘excess’ is used as a noun or a adjective; its verb is ‘exceed’ ;-)