Ooh, sounds just like my DS! I mean, exactly like my DS! Except that we didn't co-sleep. So that was my housework and me time.
Nothing really worked here, and he's just starting to grow out of it in the last couple of months (he's 3.5), to the extent that I can go to the loo with him at the door, or shower without tears. But there were a few things I did to help with my own sanity...
Organise myself so that housework took the minimum amount of time...put the washing on the timer at night so it was ready to hang out, start prepping dinner at lunch time (cos it'd take til dinner to get finished), use the hand vac as it was quieter and easy to park somewhere if I could only do a room at a time, use a clothes horse rather than a high washing line, involve Bub. Not like 'normal' babies where you can give them a bowl and cloth while you wash, a pile of pegs or their own towel to fold, but actually have them doing what you're doing - DS would sit in the clothes basket and I'd do one item at a time, pulling it on his skin as it went past, then quickly rush it on the line before he realised I wasn't touching him! I'd put the pegs on his shirt or in his hand. Worked about 20% of the time...As he got older, he'd sit or stand on my feet or hide in the washing line while I did 2 or three things at a time. I'd be able to get a bit of dinner prep done while he had arvo tea - so finger foods became really important! The highchair was touching me.
Me talking/singing (til I lost my voice!) bought a 1 minute shower before the screams started and having the bouncer in the loo with me manually bouncing it sometimes allowed me to go to the loo without him ON me! Daily baby massage and snuggle time in the arvo with my choice of music helped me stress less (almost like I was the one being massaged). If I showed I was agitated/completely and totally over being held to ransom all the time, he'd get even more clingy, so being calm helped. Eating while I held him, or while he ate was the norm - never try and eat when he's not eating or with you! And being hungry made it harder to cope with it all.
Are you happy with the sleeping? If he slept without you, would that give you enough sanity time that the rest of the Clinginess would be somewhat manageable? If so, maybe slowly work towards getting him so sleep more independently...
Bookmarks