@Patrycja we are doing this and it definitely helps. My husband also puts one of his T-shirts over the mattress so it smells like him and not like me/milk.
Seems like ur having baby blues and it’s totally normal. Just know this is a short phase. It does seem daunting. But it will slowly pass. You are doing great!
We wrap him up in a blanket whilst feeding him. Then rock him as we lower him into the crib. Place a hand on his chest and slowly release it. Seems to work - most of the time!
Something that has worked for us is we put a heat pad in the next ro me for like 2-5 mins when I'm feeding to heat up the cot so the baby doesn't feel a temp transition from your warm body to a cold mattress. Maybe that's worth a try?