air leaking around windows
Caulk is usually applied only outside. That's where airflow should be addressed. Caulk on the interior makes for an unappealing appearance and, as you suggested, treats a symptom and not the primary issue. 

I would apply interior window film, if you think the remaining winter season warrants it, and deal with your window installation issues in warmer weather. Windows installed in the last few years should've been foamed in, which creates a fantastic seal to the framing. they should also be caulked to the housewrap or wall sheathing as the situation allows. No air management should need to happen on the inner face. Window trim is not intended to be a component of the building envelope.