You know, when you've just started dating, you want to be all clingy with each other, right?
Literal
Look, dated [just-ばっかりの] period [topic-って] cling-do-want [isn't-it-じゃない]?
ばっかり is the colloquial variant of ばかり; ~たばかり means 'just did X / having only just X-ed.' The って particle is a casual topic marker here standing in for something like といったら/というのは. ベタベタする 'to be clingy' uses the mimetic ベタベタ 'sticky' as a する-verb — one of many mimetics productive in this construction. ~じゃない with rising intonation is a familiar tag question seeking agreement.