We had similar in our family. My mil was meant to look after my niece when Sil went in to have their second child, at the time mil drank a couple cans of cider most evenings but more often than not she'd drink more and get drunk, not tipsy, but merry but slurry, stumbling drunk and my sil had the same worries you did. She also was the plan for looking after the children when sil returned to work. Bil did ask if she's lay off the booze in case baby came late on, and mil got offended because it's "only a couple of cans" and there's no way she's not drinking for a few weeks.
They ended up taking my niece to hospital with them. Not ideal but it was either that or bil stay home and sil goes in on her own.
How do you know they don't drink when they have your son? My mil, unbeknown to sil and bil was actually drinking when she was looking after their eldest. A neighbour popped in after hearing crying, she got worried when it didn't stop and went to see if mil was ok. She'd passed out drunk.
I knew for a fact she was drinking during the day though, I told the relatives who received free childcare from her but they continued leaving the children with her because she said she didn't drink when she had any of the kids and that I was exaggerating because I'm a judge tee total.
I wasn't exaggerating and it took three quite serious incidents, a phone all from the school to say they are refusing to let the children leave with their gran because she's that drunk she's fallen in the playground three times for those close to her to accept that their mum can't be left alone with the kids anymore. Sil wuizzed the eldest grandson why he didn't her gran was drinking so much and poor kid said he thought it was normal which to him it was and since birth he'd see her pouring cider or gin almost every day.
It's not just going for baby number two where this could cause problems and while I hope it's never ever needed it might be best to have something worked out for other situations where they can't be relied on for childcare, like what if one of your family takes seriously ill and you want to go visit and see them but in-laws are drunk? What if your dh needed an urgent hospital visit but in-laws are pissed?
Your instincts are telling you they can't look after him when drunk. Listen to them.