I don’t expect my sister to give up everything to ok after my kids.
But you do. You've decided that she must be lying to you about having plans, and whatever her real reason is for saying no, it's not good enough. She can refuse to look after your kids, but only if you decide she has an acceptable reason. Otherwise you'll get the hump with her.
And then you wonder why she lied?
I’m pissed off because she makes no effort with me or my children and she doesn’t help at all with the care our mother needs.
I'm sure she can tell. Let me guess: you stew in resentment about this and have never attempted to have an adult conversation with her about it?
You chose to have four kids, and you're choosing to take on the bulk of care for your mother. It's human to want help, and it's understandable you feel overwhelmed. But at the end of the day, these are choices you made, and you're acting like your sister owes it to you to do everything you're doing. You resent her for being single, and constantly compare her to your husband's siblings. Do you really think that's not coming across when you talk to her? I'm not surprised she ditched the weekly dinners.
Yet she will still expect me to do things for her eg retrieve parcels from her front door. Give her a lift to airport or hep her out with a late credit card bill. And yes I feel I’ve a mug.
Then say no. Make an excuse. Give her a taste of her own medicine.
You won't, though, because you're worried other people might judge you for it. And because you want to have things over your sister, as fuel for your resentment. You need these things in your back pocket, so that next time you want her to do something for you, you can say "Well I did X and Y, you owe me". (You won't actually say it to her, but you will be thinking it. Loudly. And feeling hard done by.)
You should help your sister because you want to help - not because you want to get something out of it. The way you're doing it now, you're making the relationship transactional and feeding unhealthy resentments. Maybe your sister is an ungrateful brat. But you play the martyr and feel entitled to her time. No-one is winning here.