It sounds like you don't think I'm doing the right thing, which makes me feel better
Since he is
awake when you put him down what you are doing is just fine!! He is still little and just now starting to self sooth.
Self soothing is a learned skill, that literally takes weeks. It's not something that will happen overnight, or within a few days. Your little one is just 2.5 months old, so he's just starting the process.
If you create a sleepiness scale from 1 to 10 — 1 is full-throttle and 10 is out cold. Wait until your baby hits number 7, and lay him down to sleep. It sounds as if this is what you are doing, so you are right on track!!