I completely get how you feel because I also get hypochondria regarding cancer. My family is also incredibly pale and moley which is a risk factor. 
 

However, most pale moley people don't get skin cancer. Only 17,000 melanoma cases are diagnosed every year in a uk population of over 67,000,000. Squeemal and basal cell carcinoma are more common but rarely found in pigmented moles- they also don't often spread. Age is also the main risk factor so the vast majority of those cases are occurring in people more than twice your age.

The likelihood you have skin cancer is incredibly minimal. If you have moles that are changing then having them checked by a GP won't hurt- but you're very very low risk. I'd also add that moles change under the age of 25 as you're still growing and that's perfectly normal. I got new moles between the ages of 21-25 (my current age) that were absolutely fine. 

My mother was a sunbed lover who never wore sun cream and she's yet to get skin cancer at 50.