I know someone who has two apartments which are sometimes available for long term rent, one opposite the tennis courts in Los Gigantes and one in Sunset, in Puerto Santiago (round the corner from Drago Hotel). Send me a personal message by e-mail and I will try to put you in touch. Delmar may also have rentals. Alternatively, type "long term rent in Los Gigantes Tenerife" in Google and see what comes up. I checked and there are a few.
Regarding health care, make sure you have your EHIC card (and that it has not expired!). You can register at the clinic behind the Vigilia Park arcade, opposite the post office. You will need your EHIC card and a photocopy of your passport and the original. You may also need your NIE certificate, which you will need anyway to open a bank account, etc. The clinic is free. There is a good pharmacy (English speaking) just down the hill from the post office, on the left. They speak excellent English. I believe it is required that you speak fluent Spanish at the clinic which, for many people, means taking a translator. What I have done in the past is taken my UK prescription to the pharmacy where they will write a note in Spanish for the doctor, requesting a repeat prescription. This works. If you don't speak Spanish there is an excellent school (with classes of about six) in Callao Salvaje and again, I can give you details if you are interested.
Utility bills will obviously vary with the size of the apartment and life style. Gas (bottled) is about half the price in the UK, delivered to your door. Electricity is more expensive but you will undoubtedly use less. I think water is about the same. So not a lot of difference overall, except the wonderful climate means you use less.
Good luck with your move!