This sounds pretty similar to what i went through with my daughter at that age. She's 3.5 now and honestly, just within the last 9 months or so she's really been more accepting of her father's help! She also is outgoing, loves her extended family of course loves her Daddy to death, but she always wanted me and only me. Honestly, I think it's one of those things that you sort of have to deal with for the most part but I'm not opposed to some tough love. I would tell him that you aren't going to carry him to the store or in the store and give him choices....would you like to hold this hand or this hand while we walk in the store? Are you going to ride in the cart or walk beside me in the store?, etc., etc. If he doesn't answer, you choose for him BUT be prepared for crying, whining, disapproving looks from other people, etc. As far as the dad issue, do the same thing...Mommy is going to hold you or sit next to you (or whatever) during most of church today but when I ask you to go with Daddy you need to go with Daddy. Keep telling him that over and over...and again, expect tears, but honestly, unless you NEVER want a break, I think it's okay. I did it with my daughter and it was kind of heartbreaking b/c of course you don't want to see your kid cry or hurt them, but c'mon, Kid! ;)
I think it's partially the age, but like I said you can't be expected to be stuck to him 24/7, so do what you can and hang in there. Eventually it will get better, but mine took a looooong time, hopefully it won't be that long for you!! ;)