I know nothing of Turkish. But it's extremely feasible that the trailing vowel is decided by tense or inflection. Note that the active subject changed between the two phrases. "You run out of arguments" vs. "They are ____ you."
I studied Turkish in college, but it's been a few years.
Turkish has "vowel harmony", so a suffix's vowels are determined by the preceding vowel. The ending "-IncE" (capital = "correct-by-harmony-rules vowel here") means roughly "when doing", and it can look like ınca or ince (or other variants) depending on the last vowel of the verb it's attached to.
Literally "Zaten sen sıkışınca konuyu değiştiriyorsun" = "Anyway you (when-)get-stuck (to-the-)topic change(-now-you)". They don't quote the other reading of the sentence (or even the alternate word), so I think the "they are __ you" form was just the English translation.
It's not even clear the message used an "e" -- just that "sikişince" is what it "looked like". I propose that people, when reading, tend to place more value in the root of a word than its suffixs. He probably typed "sikişinca", which I admit looks more like "sikişince" than it does "sıkışınca".
