The only 3 player variant of Canasta that I know is Cutthroat Canasta. Are there any other good 3 player variations?
While I'm not a huge fan of this variant, everyone else in my family is and it is now the only version of canasta played regularly for us. There are a lot of slightly different rules set out there, I've linked to the one that seems to resemble the rules my family uses.
The main difference for Hand and Foot is that each player is dealt 2 Hands (called a Hand and a Foot). The player only uses the "Hand" to start. When they drain the Hand of cards, they may immediately pick up the "Foot" and continue the turn they are on. Timing the pickup of your Foot is the key to the game.
Any number of players is well supported, we play with three often. Four decks will give you a nice game of Hand and Foot for three players.
My objection to this variant is that you are really dependent on what is in your "Foot" each round. For me, it introduces a larger luck factor than I prefer to play with. This leads to more games with large differences in score which I don't find enjoyable. It's definitely worth a try to see if it is your cup of tea.