Have you tried a chart? Though I dont spank i am not against it or anything, it just seems that out of frustration it may not be the best option in the long run...
My ds has an ASD and he struggles with certain things which causes him to either meltdown or black everyone and everything out, I find that talking to him the whole time at the store really helps to keep him grounded and gives him something to focus on. I also reward the good behavior, he knows if he works hard on listening while we are there he gets to go down the toy aisles and check everything out (he never actually wants the toys, looking at them is his 'thing'), so that may be a good option while out (reward him with his favorite treat and make a huge deal out of good behavior).
At home we have a chart with chores and behaviors (be nice to his sister, no meltdown/controlled meltdown, Listening, complete quiet time/school work) and he gets to put a check at the end of the day, he really loves that and it helps him see how well he is doing. He also gets a choice to trade checks (30-hotwheels car/dollar, 40- Go out for ice cream and so on). I was very skeptical about charts as I had tried before and it had not worked, but now it is like something clicked and he is working pretty hard to get that ice cream trip.
Whatever you decide on I want to wish you the best of luck <3 dont be too hard on yourself mama, parenting really is hard and whoever said it got easier was wrong
I have never really had a bad experience with my son while doing the shopping but I have just started letting him help me. It does make the trip a bit longer but he has a fun learning experience. I started off with just fruit and vegetables where I would ask him to get the carrots he would go to the carrots and I would say I would like 6 please and he would count them as he put them in the bag and so on. I also asked him what colour things are. This could keep your son busy enough to not chuck his tantrums. He also gets some yummy morning tea for being a good boy and helping.
He normally wants fruit salad.(go figure) lol