Dont worry about looking incompetent! All bubs cry, and all are bound to have numerous meltdowns in the shops.
Rug up if it's cooler, and hopefully the fresh air will help.

Can you ask your mum when you see her during the week if she would watch her, so you can catch up on some sleep.
Get your DH to do some of the night feeds, so you can get a decent block of some sleep some nights, will make it easier if she is unsettled the next day. (she is bottle fed isn't she?)

And, you are doing the right, if it gets too much, pop her in her cot where's she's safe and go away for a few mins to regain your composure.