To enforce "exactly one of x, y is true," combine two constraints. First: at least one is true, that is x OR y. This is standard -SAT. Second: at most one is true, that is NOT x OR NOT y. Together: exactly one. You can generalize this to more variables with care.
For more than two variables, the "at least one" part needs auxiliary variables to stay in -SAT. The "at most one" part uses the pairwise pattern from before. Three variables requires more clauses.