Building a full adder with NPN BJT transistors
It's probably most straight forward using a NOR form as the basis of a BJT adder. It's quite simple to form the basic gate, this way. Nothing crazy, at all. Just simplicity.

The following schematic shows the basic one-BJT form for the NOR gate at the top. It then follows up by showing what a full adder would look like if it were based entirely on these gates. You'd need a total of 9 NPN BJTs here. Half of what you are using now. And it should actually work okay and simulate fine. (You may get glitching. But if you observe reasonable setup and hold times, it should work okay.)







simulate this circuit &ndash; Schematic created using CircuitLab

Also, the inputs as designed don't overload the outputs. There are, at most, three input loads on a single output (in two cases.) The rest is either one or two loads.



Here's an LTspice circuit and simulation, walking through all eight combinations of A, B, and C.