5m old hasn’t slept well ever - HELP

This is going to be long and ramble-y, I apologize. I haven’t slept in forever.

My baby is 5 months and a week old. He hasn’t ever slept well. I feel like I’m doing something completely wrong. I frequent this sub and read problems that other parents are having, but we’re having such basic issues I feel like I’m completely failing him.

It’s been 5 months and we still haven’t been able to establish any kind of schedule at all. My parents live 20 minutes away so I frequent their place all the time and sometimes I wonder if this back and forth maybe causes issues with his schedule? (Although I tried not commuting for a couple of weeks and nothing changed)

He has always woken up from naps crying and tired. He’ll be rubbing his eyes literally 5 minutes after a nap. We barely have any happy smiley time. His schedule is completely dependent on when he decides to wake up that day and how long and when he decides to nap. It’s different every day. I try to follow a rhythm but he’s very temperamental and it never works.

For example: we’ll wake up between 5:30-7am. Fine. I’ll feed him and sometimes he’ll go back down but sometimes he’s up so I treat that as a wake window. Within an hour he’s cranky and ready to sleep again. I’ve tried getting him to sleep at that point and I’ve also tried keeping him up for an extra hour (2h wake window for 5m olds) - but regardless he’ll sleep 20 minutes if I lay him down independently or sleep 2h if I contact nap with him.

If he wakes up in 20 minutes - he won’t go back down, stay up another hour or two and just be fussy and angry. Then I’ll try to get him sleep somehow and he’ll be fighting it the whole time. But same story, he’ll sleep 20-30m unassisted and up to 2h contact. We continue this 2-3 times until the evening when we take a bath and try to sleep.

Overnight he’ll wake up 4-5 times always crying and will self soothe maybe 1/10 times.

I’ve had people tell me he’s chronically overtired? I have no idea how to fix this. I tried doing nothing but contact napping with him for days in the hopes that I’d pay back some of his “sleep debt” but nothing helps.

He wakes up crying. Happy for a few minutes if I’m interacting with him right in his face. Crying and rubbing eyes immediately after.

I’m sure I’m not providing some useful information. If anyone was able to get through all that, I would appreciate the insight and be happy to answer any follow up questions.

I just want to do right by my baby. He’s always miserable and I’m sure I’m doing something wrong.