Hi OP I clicked on this wondering whether you were having similar issues to me (which I think you are but my DD is younger) and hoping for some advice too! You are not alone.. rather than start another similar thread I'll describe what my DD is doing and what I've tried..
Basically she gets really eager to go out for a 'walk', demands it. We'll go out, then when we are a certain distance from the house, she'll demand to be picked up.
She's just gone 2 and I'm not very big just over 5ft, so she's already getting quite heavy for me to carry.
The real problem is a day like today.. we go out with her dolly in the toy pushchair, she happily gets to our destination. I pick up some shopping so now have a shopping bag. Then as soon as we start to head home, she wants to be carried. When I say no but you can hold my hand, she sits on the floor and no amount of cajoling, scolding or anything will move her. She will migrate from sitting, to lying down, on a soaking wet pavement. She has a lot of endurance for staying there (half an hour) and looking pitiful/wailing to get my attention and passers by attention.
I end up caving to her, partly because I'm so embarrassed and doing my best to carry her, the shopping and the pushchair/baby, even uphill, by shifting her from one hip to the other, but it takes all my strength.
I've tried holding out once and she did eventually get up and walk. It did take half an hour and I was really cold by the time we started moving again.
Passers by either look/comment in an 'oh, I feel for you, we've all been there' fashion, or glare/make some comment. I had one woman out dog walking actually pick her up and bring her to me when I had just walked a few paces away saying look how dirty she is.. which wasn't helpful, and she was in a snowsuit and wellies that day so there was only so dirty she could get stupid woman bet she'd have a fit if I picked up her dog
The advice I've had from parents with say, 4 children is that they would just leave the child there and hold out until they get up and learn that you are not going to pick them up.. In your case, you can't not go anywhere, or practice this when you are in a hurry, but maybe there is somewhere (park?) you could go for a few days running, when not in a rush, but pretend you are going somewhere, then let him do his thing on the ground, don't give in to it, see how long it takes, use positive encouragement the rest of the time - kind of super nanny style? I'm suggesting this because it's what I need to try before DD gets any bigger.
If anyone else has any suggestions, happy to hear them!