That should fail in the long run because it's unlikely that the frequency of letters in English divides so evenly that those rules make each choice converge to happening exactly 1/3 of the time.

I'd just generate the random numbers in my head. A useful thing to do is to pick a couple of numbers from thin air (which doesn't work by itself because the human mind isn't good at picking 'random' numbers from thin air), then adding them together and then taking the last digit (or if you wantt 3 choices, taking them mod 3).