Hello.

A few times I have seen verb forms like yapasım or unutasım in use.

Could anyone explain their meaning and grammatical origin, I mean - what kind of suffixes they consist of?

Is that yap + as (if such a suffix exists at all) + possessive -ım

Or perhaps yap + a (like in the participle güle güle) + s(a) (reduced conditional suffix) + possessive?

It is ası/esi suffix.  This suffix makes adjectives from verbs.You can use these adjectives as nouns too. This suffix also gives the verb a wish or recommendation meaning.

Ex: Görülesi yer. A place to see (you wish to see it or you recommend it to someone else)

     Geberesi adam A man you wish him to die.