Peace and thanksgiving offerings were shared by the worshippers. In one sense the worshipper was giving up something from his flock, but in another he was sharing a meal with God--not exactly giving up anything.(*) In fact a few commentators think these were some of the rare times when the ancient Hebrews ate meat. Think "party times." Notice what was prescribed for the tithe. That would have been a big feast.
Thanksgiving is kin to praise. Peace is related to the right ordering of things. I gather that the peace offering could be free-will, along with a vow you fulfilled, or for thanks for your deliverance. All seem to fit with "right ordering."
Psalms 22:3 is a bit vague, but modern Hebrew experts(**) think it doesn't mean that God inhabits the praises of His people--though Jesus said where 2 or 3 were gathered in His name, He was among them. So maybe it does mean that after all.
Songs of praise, thanks, and celebration of "right order" seem to fit with a few of the aspects of OT sacrifice: worshippers, enjoying the "sacrifice" together with God, and, of course thanksgiving or peace. There's no physical "giving up" something, but as with the ram and Isaac, we don't always provide the material for the sacrifice.
Just don't sing by rote while your mind is chasing squirrels. Like mine does.
(*) Ayn Rand was a fool, probably because she didn't look honestly at herself. Sacrifice of things isn't giving up a greater value for a lesser, but the opposite. The reason we encourage it is because people (Rand included) aren't always eager to act rationally. If you want to earn yourself a STEM degree, you are almost certainly going to have to sacrifice your habit of late night parties. Some people don't, and generally don't get that degree.
(**) And ancient experts: the Septuagint translators seemed to think that it meant that God, the praise of Israel, dwells in a sanctuary.