I'm 69 and have been SUPing for two years. A year ago I had my right shoulder replaced, now my left one is acting up. It would help if I did' have to switch sides to go in a straight line. Is there a paddle stroke one one side that will keep me going in a straight line?

No, but if you were in a row boat with just one oar, you could do a skulling stroke off the stern, keeping the blade in the water and doing a figure '8' with it, with a bit of wrist action thrown in.
Other than that, even a radical 'J' stroke can only work for so long.

I think it would be possible LBP. When in Milne Bay outrigger canoes could be paddled on one side. In fact, in some areas the outriggers were covered in and one had to be able to paddle and steer from the one side. Admittedly they had had years of practice - even the young kids! I'm going to try paddling on one side tomorrow and see how far I can travel in a relatively straight line. (I have paddled the outriggers with some difficulty especially if it was windy).

Sounds like a recipe for customising the biggest / widest fin in the history of the world to me!

To go straight, here are my 3 tricks:
- picture the "nail" www.gong-galaxy.com/forum/viewtopic.php?p=43629
- keep your shaft in a vertical plane www.gongsup.com/forum/viewtopic.php?p=77774#p77774
- open you blade at the catch (less tiring than a J stroke):

I have found that you can also use your weight. Move your weight to your preferred side of the board to bury the rail and paddle on that same side. You will find you will go straighter for longer before switching. This method is slower through the water though and the leg you place more weight on will get tired quicker than your other leg depending on your fitness.

Or just get a board with a rudder.

Also I seem to recall Jamie Mitchell had a fin advertised with a one side bias in it.

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Snowie said..
Or just get a board with a rudder.

Also I seem to recall Jamie Mitchell had a fin advertised with a one side bias in it.

This is probably the best solution, but it can be expensive.
But anything you try will slow you down, your paddling will be compensating (fighting against) whichever solution you try, producing increased drag.
Increased drag is harder on the shoulders than paddling with minimum drag, something to seriously consider.