Get a supply of the plastic clips, like a clipboard clip but smaller and double-stick tape on one side. Put them around in handy places, e.g., where one can hold the toothbrush for you.

Teethbreesh: hold tube backward, cap between thumb and forefinger, body of tube in palm of hand. If you’re dextrous enough to type, you’re dextrous enough to get the cap off from that position. Out-of-the-box thought: throw the cap away. Why does the toothpaste tube need a cap? Sure, the first thirty-second of an inch of toothpaste will dry out a little. But cooties will get in there! Sorry. If the germs able to colonize a near-sterile, antibacterial substance overnight can cause you problems, the nurses who maintain your hypersterile plastic bubble can open the toothpaste for you.

Regards,
Ric
[other one-hand tips upon request; loss of use of a hand is an often-encountered temporary experience for me]