No, you can't get mercury poisoning from a flu shot.
Some people do experience allergic reactions to the flu shot and people who have egg allergies should not have the flu shot. Some people may even come down with flu like symptoms after having the shot for a few days.
Having catarrh in the throat can sometimes taste metallic and also if you had any bleeding in the mouth from your gums. Blood tastes metallic.
If you have had your fillings for a very long time, it is wise to see a dentist to check your teeth. Mercury fillings can break down and leak out. The dentist will let you know if any of those 3 metal filled teeth need restoration and also examine your other teeth and gums. Some dentists also screen for mouth cancers. Should you require replacement fillings, opt for the white ones or Cerek.
Brush your teeth twice a day and floss your gums. Flossing will help to strengthen your gums. If you do not floss, you may have some gum bleeding when you first start, but this will stop as your gums get healthier with the flossing.
As you have been feeling unwell since September, I suggest you make an appointment to see your doctor with a list of your symptoms. It is likely that you may have a viral or bacterial infection. If so, your doctor will prescribe antibiotics for bacterial infections. Viral infections do not respond to antibiotics.
The doctor may decide to order bloods tests which he may feel are required, which will depend on what your symptoms are.