M.G.
I was lucky and so long as mine could see me, they were content.
Do you have places to plop him? I had a place to plop in every room because I simply couldn't carry a baby around all the time. I had a little area safe for baby, set up in each room. As we made way around house, I'd plop them there (totally supervised) but I got stuff done. So I had an exersaucer in one room, a play mat with toys in another, etc.
That might help to keep him interested in something other than you so you can at least get something done.
If not, then I say go for the carrier/sling idea. My husband would have been on board with whatever I needed to stay sane and keep babies happy. My husband knew I had more knowledge of the baby's needs, etc. so deferred to me on that.
A friend of mine had a velcro baby (love that term Diane). It was hard - she thought it was something she was doing, until she had her second and he wasn't like that at all. So likely he needs comfort. I would just keep gradually keep trying to get him to do things like sit in his seat - how about putting him in a highchair and have toys on the tray? And then stick some Cheerios on it so he gets used to enjoying highchair and gets a snack in?
Same for shopping cart - maybe have a new fun toy that only comes out at grocery store that attaches to cart ..