Convert (x + y = z) to propositional logic formula for 8-bit numbers
You can obtain a formula in CNF form by writing down a truth table (with $2^{24}$ rows) and then converting each row to a single clause.  This will yield a formula with $2^{24}$ clauses, but the method is straightforward.

If you want a shorter formula, I suggest using logic synthesis, e.g., Quine-McClusky, Espresso.

The usual path to get a small formula is to allow introduction of additional variables, build a circuit (e.g., an adder circuit followed by a comparator), then convert to a formula using the Tseitin transform.  This yields much smaller formulas, but it requires introducing extra variables, which you said you don't want.