If you’re looking to go to the beach and such, Benalmádena is a better option (and probably cheaper) but there won’t be too much to do apart from going to the beach unless you make day trips to other places like Málaga, Nerja or Marbella.
If you prefer staying in the city, Malaga is a better option. There are lot more to do than just museums. At Malaga, you can find many hotels.My personal favorite is Alcazaba Premium Hostel for its rooftop bar and clean rooms at affordable tariff. The other hotel you must consider is Hotel Finca which is on the outskirts of Malaga.
I agree with you, short-term apartments might be better than a hotel. However, not aware of any local websites that list apartments. Probably you could look in www.milanuncios.com, which is a website for personal ads, and there might be some options for you there. Another option is www.homeexchange.com where you can exchange houses with someone that might want to visit the place where you live. But if that’s not something you’re looking for, there are many good options on Airbnb and there shouldn’t be any problem with rental scams as long as you make sure you don’t pay outside the platform.
If you are looking for apartments, choose one in Old Town area. The area is central, atmospheric with historical buildings, shops, and a thriving street-café culture. If not the old town, you can also look for apartments in the nearby Los Montes district.