Hondas H22A engine burning excessive oil?

Engine overheated and now it`s burning oil. I know it`s because of overheating. But can`t really come up with what could be the problems.

I`m probably going to take off the head and take it to shop for measuring if it`s twisted. And changing gaskets. But what else could be the problem ?

Also, exhaust gasses are white. They`re not blue as in burning oil. But I think it does smell like burnt oil. Not really sure, but the smell is different (it actually started to smell when temperatures dropped below zero celsius outside). Haven`t driven it a lot because other problems I have to fix, but I think oil is being burnt more now when it`s cold outside.

Could it be coming out through gaskets and getting on exhaust and the smell comes from that ?

Any ideas will be appreciated :) Engine is H22A. 2.2 liter from prelude (BB5 or BB6 probably). It`s swapped in Accord CG4. Not that i matters :D But still.