My English Springer Spaniel puppy is 13 weeks old. She is very submissive and when I try training techniques she lays down or turns away. She does 'come' and 'sit' but when I try anything else I can't keep her interest. Is she too young? When do I start more training? Also, I know socializing is important but the vet stressed that she is still susceptible to disease until she has completed her shots. She still has several more shots to get. Do I wait to socialize or will that be too late?

It is not too early to train. You can start on the basic training (sit, down, wait, stay, leave it, drop it etc) and also getting accustomed to a leash. Socialization is key here. She should be introduced to all kind of environment and people if possible. You have to be careful because she hasn't taken her shots. E.g. when you take her to vet, carry her and keep her in your lap and when the vet checks her, ask the Vet to give her a cookie. Carry her to friends house. Don't allow her to walk or run outside as of now. If possible, enroll her in a puppy socialization class. You will want to check the class and see if they are keeping it clean and have some requirements on vaccinations.