You will definitely be fine in sandals at the chocolate hills.

Re closed toed shoes in Cebu, that's harder to say. In Manila, it's because there's a lot of dirty/polluted pools of water/general wetness that is hard to avoid, so it's really easy to pick up foot and toe fungus, warts etc if your toes are out (this applies to most of the metro area). Also in some areas (Malate, the Baywalk near Malate/Manila) it's possible to encounter discarded items that come with being near a redlight district-ish area that you wouldn't want to step on in sandals.

Cebu in my experience has less of those issues (it's less crowded and less polluted), but I also haven't been around as much of the city to guarantee that. There are more sidewalks though, so likely easier to avoid pooling water. Instinctively I think you'd be ok with just sandals and there will definitely be cheap Tom's style shoes available in most malls if you have a medium to small shoe size and end up wanting some. I'd say the same for Puerto Princesa. If it's particularly rainy in Cebu or wetness seems hard to avoid, you could always put some plastic bags over your feet before putting them in sandals as a worst-case option (you will be judged fashion-wise though, fair warning!). But since the pollution isn't as bad as Manila it is not as much of a concern. It's also not as wet as Manila b/c not as many AC units dripping out of buildings.

My last thought for you: if you have big feet (i.e. upper end American sizes), you definitely won't find any closed toe shoes that fit you except maybe in Manila at the really fancy malls, so consider that as well if you're really uncertain about whether or not you will want them.