We didn't swaddle him but have begun doing it.
It's actually dangerous and an increased SIDs risk to start swaddling after 3 months old. It is fine to swaddle a child used to being swaddled from when newborn. The danger lies with introducing swaddling late.
I would suggest a very tightly tucked in blanket or sheet instead. Not quite as effective but does have a smiliar effect.
he doesn't know how to self soothe
His dummy will do the soothing, so don't worry. Longer term (7-8m ish) he'll have the dexterity to put his own dummy in. So independent soothing will come.
thrash his head from side to side and grunt
These are absolutely classic "I am exhausted pleaseeeeee please PLEASE help me get to sleep RIGHT NOW". It suggests an over tired baby.
I'd work on more daytime naps. Any sleep anyhow. Movement helps - bouncy chair when home, pushchair walks, car rides, rocking in your arms if it's the only thing that works.
Short 20-40 minute naps are normal. ✔
Make them frequent ✔✔✔
Try to limit awake time (between waking from one nap and going to sleep the next) to about 40-60 minutes at this age.
So your day might be repeated cycles of:
🔴Wake - note the time
🔴Feed
🔴Cuddle/play - This might just be 10 or 15 minutes
🔴Nappy check
🔴Settle to sleep (I favour foot bouncing 're bouncy chair (minus play arch ) at this age
Sleep
🔴Wake.... and back to the beginning to repeat the whole process over and over again.