Second third or fourth what's already been said- pull the fabric out and use the mulch back in garden beds. Or under shrubs, surely you have someplace for it. I don't think a mulching mower will do much with it. It can go in the grass if you're easy on it, not thick.
(Commercial landscapers rake off mulch and put down new mulch in shopping centers, and I want to beg the guys for the old mulch AKA compost)