"I love you Ayse, I cannot go!"
Translation:Seni seviyorum Ayşe, gidemem!
For me it is making sense. A person (here seni-you) love Ayse, because of that reason this person cannot go (cannot leave her)
You are correct;
The definite direct object can move anywhere before the verb. It is recognised by the accusative case suffix.
"Gidemem," using the aorist ending, seems to mean (to me, anyway) that Ayshe's lover can never, ever leave under any circumstances. But what if the progressive ending were used (gidemiyorum, perhaps)? Would the meaning be more like, "I cannot leave right now (but I might tomorrow)"?