A couple weeks ago my mother was carrying a cup of coffee and plopped into her recliner dumping over her chest burning herself. She is a severe diabetic, is supposed to use a walker and I made sure she had a travel mug with lid but doesn't always use Walker, doesn't use the arms on chair to sit and continues to use mugs. I got her another travel mug and told her if I found a dirty coffee cup in the dishwasher next time I visited I would remove all cups. She has a basket on her walker she can put her travel mug in also. This am I watched her carry a full cup of coffee to the table, no walker and very wobbly. Do I make good on my threat and remove the coffee cups? She gets no visitors and if she does she can use styrofoam cups which she can't reach but her guests could. I am at my wits end, she will burn herself again and call me, I live an hour away.
I dislike travel mugs, in fact I threw away all the freebies I got over the years. They are hard to place under an automatic coffee brewer, and the tops are hard to figure out how to open and close. And hard to clean. If I was an elderly person, I wouldn't be a happy camper if that was my only way I could drink coffee.
Hopefully Mom has a caregiver who comes in during the day to help her since you mentioned that Mom is a severe diabetic, has memory issues, and is unstable on the feet. She shouldn't be left on her own.
Get Mom a tall china mug, less chance of a spill unless she falls. These tall mugs can be put into the dishwasher instead of filling up a trash can with Styrofoam cups. Guest would like them much better, too. Styrofoam cups can get pretty hot to the touch, unless Mom has those cardboard circles that go around the cups to keep hands from getting burned.