Were Saul, David, and Solomon qualified to offer sacrifices for the people in the Temple? I thought that was the work of the Priest.

Donating a sacrifice to be brought at the temple for G-d can be done by all. The only ones who can offer the sacrifice are the Cohanim - the priests. One of the stages in the offering can also be done by somebody who is not a priest and that is the Schita - the actual slaughter. Rabbi Ro'i Margalit