Maybe it's just me, but I keep mixing up As in
I've completely forgot
I haven't the faintest idea / I really don't know
are there other expressions to use them?
Note: This is not a direct answer to your request for further usage examples of the two terms in different verbs. I rather try to explain the nuance with the help of a diagram and a pair of contrasting examples for each of the verb you gave.
As you state in the question, both すっかり and さっぱり is about the completeness of something. The difference is that they deal with different kinds of completeness:
すっかり is interested in the degree, or progress regarding completeness.
さっぱり's interest, on the other hand, is how completely clean it is of something.
As for すっかり忘れた - すっかり is an adverb that simply means "completely, without leaving anything left out". Though すっかり忘れた is probably the most common usage, here are other examples: 「宿題はすっかり終わった」 "I completely finished my homework." 「すっかり春になった」 "It's completely become Spring."
As for さっぱり分からない - さっぱり is an adverb that means clean or refreshed/refreshing, and can also by extension mean "completely (not)" (hence さっぱり分からない). さっぱりした would mean basically, "to feel refreshed". Other examples are: 「風呂に入ってさっぱりする」 "I feel refreshed after entering the bath.", 「さっぱり売れない」 "I can't sell anything.", 「さっぱりと諦める」 "I've completely given up." etc. If you come across a construction such as ～はさっぱりだ. It's usually a bad connotation meaning "nothing good". 「成績はさっぱりだ」 "There's nothing good at all about my grades."