Morning carer should be dealing with wet sheets, which should not exist anyway if catheter functioning properly - contact the incontinence service or district nursing service (one of which will already be involved if she has catheter) and tell them things not working properly.
Has she got a lifeline pendant/wristband? If she has you can reasonably take one call from her each day and not answer the others as you know she is covered by the lifeline for emergencies. If she has not got one, then she needs one!
Will she WhatsApp? - chats via this are less intrusive and help keep her connected.
How about online deliveries, especially bulk items like bog roll. Delivery people will take into the house and unpack and that is another person for a few minute's company for her.
Talk to the carers and make a list of what they must do: sort out any wet sheets, empty catheter bag, talk to her about any supplies that are running short and make a list for you, and any other task that would help if they did it. Honestly sometimes they need a list as dashing about from one person to another tends to switch off the imagination a bit! If she is not letting them do stuff with her, you need to have a list and just tell her that this what the carers are going to do - BECAUSE YOU ARE NOT ABLE TO DO THEM AS YOU HAVE SMALL CHILDREN TO LOOK AFTER!
You have to accept that some things she would like done won't be done - or not right away. I am widowed and have a disability and sometimes I drop something on the floor and struggle to pick it up - so .... it has to wait till someone else is there .... the world keeps on turning in spite of this inconvenience. I do however have a grabber in every room - maybe she needs that?
Research community transport in your area for appointments (hair, doctor etc.) etc. and make that the default option whenever one arises. Just say you can't do it. Or she can take a taxi - I do this when I have to - goodbye savings, but what the heck else are they for!?
Talk to siblings and say: from 12th - 19th (or whatever) I am taking a week off so one of you will have to come down and stay and take over. My DD who lives further away sometimes comes down to stay for a few days and works from here online in my spare bedroom.
I am not very old and am totally peed off with the hand I have been dealt - dead OH and lots of physical buggerations - but central to my thinking is that my lovely DDs should not feel the burden too much - clearly I can't wipe that out completely as they know that I am on my own and things are hard for me and they worry about me, but I have never rung them to ask for help.... not once. They visit, and when they are here ask if there is anything they can do.
I sometimes have ready meals delivered - would she stick one in the microwave? - or a carer could do it for her.
Research local voluntary agencies - she will baulk at it at first, but if you reduce your availability she may come to appreciate a visit from someone else.
I know how your Mum feels and I know how you feel - both are in a difficult situation, but the balance is haywire at the moment and only you can change that. I know its hard but it must be done.