EU citizens with a healthcard do have access to free healthcare in Malta. If you want to be seen more quickly I would recommend going private.
You can read more on the Healthcare in Malta page.
Hi, I'm sure by now you have already moved with your family, and whether you have or not also found out about the healthcare system in Malta here are a few pointers that you might find helpful.
The best thing to do is to apply for your ID card in Valletta, and then via your EHIC card apply for your Health Entitlement Certificate also in Valletta, also to obtain certain benefits as an older person you will also need to apply for Residency in Valletta. Once you have these you can go to your nearest Polyclinic and register as a new patient. If you have any specific health conditions you need to declare these to the doctor and ask if you would be eligible for the Yellow Paper that entitles you to free medication. If you are you will need to be seen by a specific consultant for each condition as they have to refer you to St Luke's Hospital for the Yellow Paper. St Luke's Hospital is in Gwardamangia. If you have any disabilities it is recommended to register for these at KNPD (Disabilities Commission), where you can also apply for a blue badge and a special yellow card that gives you some extra help. If you are bringing a car into the country and you are already registered disabled you may get help with duty on entering the country, this is based on age of the vehicle, emissions and other things. If you plan to buy a car here, still register with the KNPD because you can be given discount on the road tax.
Electricity and Water is the next point - if you are living in rented accommodation Landlords have to allow you to approach ARMS Ltd electric company to make sure your electricity is charged at the accurate tariff. If your bill states 0 and there are two of you living there you will get a reduction in the charges. If you have bought property the same applies. If you intend living with family - make sure they know of this rule too.
I hope you have already found out this info, and if not some of what I have passed on may be useful.
Just one more point about taxation - there are some good taxation people on the Island, I personally use Aldo Vassallo in Msida, he sorts out the tax and is very good with advising you what best to do. It is now unlikely that you will pay tax in Malta as personal allowances have increased sufficiently to help with this.