For primary stay at home parents- how do you split the time caring for baby when the other parent is home?

Looking for advice particularly from couples where the working parent isnt in a typical 9-5, but open to all perspectives! This is probably most relevant for small babies too, as my boy is not mobile and needs hands on time all day.

I am a stay at home mum for our 5 month old, he is a real Velcro baby so only contact naps, but also is very curious and gets bored easily, so every wake window I am in full entertainment mode to keep him occupied. He cries if I try to put him down even to eat lunch so I cant really just potter around and do things with him, maybe he will tolerate 10 mins being pushed around the house in the pram while i do the dishes etc.

My partner WFH running his own business, some days he is really busy and some he is super quiet and just does whatever he wants- video games, TV, exercise, sleep (dont even get me started on the day naps when I am up a minimum 3 times every night). Because of the fluctuations, he works 7 days as well.

Generally he is available if I need him to hold baby while I eat lunch, go to the toilet etc. Or entertain baby if I need to duck out for like a drs appointment quickly. Baby is exclusively breastfed and will not take a bottle, so its not like I can be gone for long anyway, or baby is with me if im out majority of the time anyway.

However now 5 months in I am really feeling the weight of being "on" for like, 23 hours 7 days a week. When he naps, I am having to use that time to order groceries, do any shopping or prep or booking appointments for baby. Husband will normally take baby for 1 nap a day (30mins) in the evening but normally this is when its my turn to cook, so it hardly seems like a trade off in my favour - of course id rather cuddle baby while he sleeps than cook dinner and do dishes. I normally have a quick shower while he baths baby.

I guess what's making it hard is every day is unpredictable for his work- i cant just say "ok at 3pm take baby so I can have a shower" and my organised brain feels like every day im chasing my tail wondering if im going to have enough time to shower or eat. Same with weekends - he might have a busy Sunday then nothing Monday, so i dont think we can make an agreement such as "on Sundays ill take baby the entire time between feeds"

So onto my question - for those either in similar or even different situations, do you have a plan or schedule for your "off" time? For couples where one works 9-5 in the week, do you have an agreement for how time is split in the evenings or weekend? How much "off" time is reasonable per week?

Also, its not like im looking to do anything in particular - id just like time to shower without rushing or water my plants without a baby on my hip trying to pull the watering can from me. And explicitly I do not want "free time" to go and do more chores! I actually would love to be able to like, go and get my hair done or go shopping, but with the breastfed and contact nap situations, I feel like they are impossible, so it really isn't anything too extravagant.

I feel like this current wishy washy is not sustainable but I dont know where to start with recommending a plan (or if a "plan" is even going to work - i feel like probably not).

Okay thank you, all advice appreciated!