Following up on Akhil's plan, a good source of projective modules which are not free is via nontrivial vector bundles: smooth sections of the bundle form a projective module over the ring R of smooth functions, but a non-free module if the bundle is nontrivial.
So, you could do something simple-minded like this: let P be the module of sections of the tangent bundle of RP^2; put a commutative algebra structure on M = R + P where all products of elements of P are zero. As an R-module, M is not free. Let N be the module of sections of the exterior algebra bundle of the normal bundle to an embedding of RP^2 in R^4. The Whitney sum of the tangent bundle and the normal bundle is trivial; therefore the algebra M x N forms a free R-module, whose center M is not free.