Hotels are usually happy to keep your luggage safe for you after you've checked out. The only drawback of this is that if you plan to go out for the day you have to go back to the hotel later, although if it's a nice hotel they'll usually have a decent ground floor restroom that you can use to clean yourself up or get changed in.
Don't worry about being on your own, as long as you're careful, don't openly flaunt any signs of wealth and keep your eyes open Manila's no worse than any other major city
