I've found and implemented a full adder to be able to do unsigned or two's complement addition on wikipedia: https://en.wikipedia.org/wiki/Adder_(electronics)#Full_adder
However, in my particular case, I'd like to use a sign bit instead of twos complement, due to there being a variable (unbounded) number of bits involved.
How can i do addition and subtraction with binary numbers using a sign bit? (say, it's a separate bit, not one of the binary digits per se).
I also only have access to AND and XOR.
Edit: Potatoswatter suggested that twos complement is still able to be done with an unbounded number of bits. What do you do when you want to add 6 bits and 8 bits together? Also, is multiply, divide and modulo easily done in twos complement? Keep in mind i have only AND and XOR, so don't have the ability to do if statements, the logic needs to be "branchless".