Hello and welcome to TP. This site is great for saying things as they really are, nobody judges or thinks badly of you as we have all been there or understand how you feel. Hugs to you as you sound like you are worn out and maybe depressed with everything getting on top of you, I understand the irony of the prison comment. Your sister has already made herself I'll by looking after your Mum so common sense says don't let yourself do the same thing.
Really the first thing I would suggest is to visit your GP and make them aware of how you are and what you are dealing with and accept any offers of help from whoever offers it. Have you had a carers assessment carried out, or an assessment for your mums needs, as this would maybe help as she may be offered day centre visit, a few hours of a befriended to visit so it gives you a few free hours to do what you need for you.
Is she safe to be left on her own, does she sleep ok, and have you thought about looking at care homes even though you may not want to consider it yet? Nobody wants to put their elders in a home but if it means it prevents you getting ill and are able to cope better mull it over a while and get your head round it.
Does your mum get attendance allowance as this could pay for her to have carers in to help you, even if it's not carers role maybe a cleaner to help you out? Have you been in touch with your local carers group, they often are supportive and you can take her with you so you both get a break.
Her anxiety when you leave her, this shows by her being moody upon your return, will increase as she gets worse, and sometimes the sufferer will follow you from room to room and you're never allowed any privacy. Try and speak to social services so that you are on their radar before you really need them ie when an emergency ever happens, hopefully it won't.
Try not to forget your sister in all this and she is one person who will understand your concerns etc but remember you need to look out and after yourself so that you can help your mum and sister, so speak to your GP ASAP. Hope you have a good day tomorrow, and keep in touch.