It sounds like you just need to reassure Eve that she is important to you. Your mother does require a lot of care and it is ok that you have to provide it. But when it comes to who you put first in your life, it should be Eve.
In any relationship, the partners should put each other first. That does not mean you need to neglect your mother. But it does mean that you need to find ways to let Eve know you care, even if it's while you care for your mother.
You can try things like making special plans with Eve where it is just the two of you, include Eve in caring for your mother, take time to talk to Eve often about how she is and what things you both can enjoy together, and make special time each week to be together, even if you need to hire someone to care for your mom for a short while.
This will help Eve feel she is important to you.
It sounds like you are on the right track, Marvin. That is exactly what you should do with Eve. Just be sure to let her know you want to put her first. Your mother's care does demand a lot of your time, but letting Eve know she is first will let her feel less on the back burner and more in your life.
Exactly! That is going to keep your relationship going.