First off, congrats for adopting a shelter dog! This little problem CAN go away, you just need 3 things:
Patience
Consistency
Timing

Patience is obvious, because this won't change overnight. you'll have little advances then she'll backslide for a bit. Keep doing things right, and she'll eventually learn where potty is to happen.

Consistency. Well, whatever course of action works, you must do the same thing the same way every day. That not only goes for you, but your husband and children as well. EVERYONE in the house needs to be consistent.

Timing is important on many levels. The obvious is what times you take her out. Putting a dog on both a feeding and potty schedule is the key to potty training. If you don't know when she last ate or drank, how are you to know when she'll need to potty? Timing also works in the manner of "how fast you react to her" In other words.... don't correct her if the pee is more than a few seconds old. Correct her right then and there. PRAISE her like she walked on water when she does it right.

By correction, I don't mean rub her nose in it to yell at her, correction means calmly letting her know potty goes outside. Even if she misses the outside, don't yell or get upset, just say "ooops, we do this outside sweety" and take her to where she needs to go. Take her despite having just done it inside. It just might work on her because you mentioned she's a bit meek. Over time, she'll understand that peeing happens outside. Will it happen tomorrow? Nope. but perhaps she'll be dependable a couple months from now.

Whatever you do, when she submissive pees, do NOT make any mention of it. If you make a big deal of submissive peeing while she pees, you'll only make the submissive peeing worse. Just simply clean it up once she calms down. Don't correct her or anything. Just clean it up.