Wear long pants today. Your knees will thank you for this. You'd think that using one of the mats at the gym would be helpful. It's not. It makes you unstable. You'll miss reps. Missing reps causes you to miss out on benefits. Plus we holler at you for dropping implements on the mats. Is NO BUENO!
You might find it's best to do TGUs without shoes.
You may do all reps to the R then start with the L if you like. You may also alternate L & R.
Don't be afraid of using a bar. Remember, the point here is to increase the strength and stability of your core. You might not be able to handle the same weight with a bar, but you'll likely add more strength to your core.
Might consider using a KB for the weight also. I know people who will swear Turkish Get Ups with a KB cured their shoulder issues.