First of all we need to correct something:
Yes, "they (de and da) are used after verb, but they are not attached with verb.
' gözlerimi açsanda ' is not true... Gözlerimi açsan da... is true...
"yanmam gönlüm yansada" is not true... Yanmam gönlüm yansa da... is true
(you can see "gözlerimi açsanda, yanmam gönlüm yansada..." some of Turkish sites, but they are not true as gramatically, believe in me, because this is a deeply grammar subject so many Turks make this mistake)
This is the first part of the answer... And the second part of it... What is the meaning of them?
"De" and "da" are short for "dahi"... [Dahi is a adverb in here, please don't mix it with the noun. Because we have also a noun word as "dahi" ]
dahi ( or we can say "bile") means "even, also"
yanmam gönlüm yansa da... I don't "care" even if my heart fall in love...
yanmak is used for metaphor in here... "yanmak: fall in love"
Shortly : da and de is equal ακόμη και, καν, ομοίως