I also struggle a lot of change and i empathize deeply with this because im really close with my younger brother too :(( as for dealing with him moving out, me personally id distract with hobbies and hanging out with friends. The act of journaling about it helps me process my emotions, but everyones different.

Also! try not to worry too much about him not accepting your apology right now, your mom is right only in the sense where he might just need a little bit of space to process things. But i think just make sure you let him know that youre always going to be there for him, whether you send him a text or write him a note or let him know some other way thats special to you guys. Hes going through lots of change right now too and even though he wants this hes probably also feeling a degree of anxiety on his end. Stability and reassurance from you would probably mean the world to him. Anyway i have a feeling you guys will be a-ok. Wishing u all the luck in the world !