Tucson Mountain Park - Gilbert Ray campground is our absolute favorite. No reservations and a 7-night maximum but folks move in and out daily so sites usually available, especially if you can arrive around 10am. This is near the Desert Museum (a must-see outdoor complex - go early morning in April). It's also near Saguaro Nat'l Park - West and Old Tucson Studios (movie set).
Catalina State Park would be our second choice. It's a little north of town so driving to see things will take a little longer.
In April a lot of snowbirders will have been heading north so you chances are good for finding something.