Your kitty needs a vet check up, including blood work and a urinalysis. Please don't even consider dumping her. Pets are not disposable objects to be gotten rid of when they have problems. She's an old lady and needs her home and people and special consideration in her last years.

She could be constipated with megacolon, she could have kidney failure and/or a UTI, she could be diabetic, or developing hyperthyroid. All these things can cause litter box issues. In the case of constipation and UTI she will blame the box for her pain and trouble, and so avoid it and go elsewhere in the house.

Arthritis can also cause these kinds of problems. The box is too far away, or to difficult to get in and out of. It hurts her to squat, so again, she blames the box. Where is her litter box kept? Keeping a low sided box in every room will help her be able to get to the box in time.

She could be losing her sight and/or hearing and becoming confused.

Please realize that your senior cat needs special care and understanding now. Seeing the vet a few months ago is not enough, a senior cat's health can change from one day to the next. And these problems she is having now are a red flag that something is not right with her.