It is a good thing she isn't overly barky and a very good thing that she isn't growling at other dogs. She is still a young puppy and with socialization and familiarity with your home and family her confidence should grow.
Since she is confident in you let her see that you are confident and calm in the situations she finds scary. Let her see that you are affectionate with your husband and interact with him and have control of the situation. Has your husband tried to engage the puppy with treats yet? Who feeds her? If it is always you that feeds her maybe try letting your husband feed her half the time so she can see that good things come from him also.
Vets are scary places for puppies. Other dogs can be scary for puppies. This is all normal. Once she is fully vaccinated (and not until then) start taking her places with you. If you are going out for coffee take her with and drink your coffee outside and have her sit at your feet. Feed her small treats for good behavior and calmness.
Socialization and a calm and stable leader should help her gain confidence. As for the barking count yourself lucky if she never starts. A lack of barking doesn't mean a lack of confidence. In many cases it can mean exactly the opposite.
Nik's right. Especially about the barking. When we went to see our pup, I did hear her bark...once. She was three months old and was calling, for lack of a better word, her brother that was over by the back fence. When we took her home, she was silent...quiet...hardly made any noise, except in her dreams, little wimpers or tiny little growls. I was a little worried...she was so placid.
I used to take her to my daughter's house and she would sit in the window with her Rottweiler watching the street. Bella would be barking and barking at people outside and Bailey would just sit there next to her. But one day when Bella was taking a break to get her breath, out comes one woof. And that was it, after that, every time we visited, Bella would bark and when she'd stop, Bailey would give out the single "woof." So Cute.
When she was old enough, we took her to the dog park, and on one occasion when she was about 7 or 8 months old (?) we encountered a Giant Schnauzer who just kept barking and barking at her. She took it for a long time just starring at this large noisy dog, and then she just gave out one grown-up sounding bark and it was loud and authoritative and that Giant Schnauzer shut up. Yep, she found her voice. I was elated. And people told me that I'd be sorry, that she'd drive me nuts barking, but they were wrong. She barks when someone is at the door, she barks when she sees someone near the house and she doesn't recognize them. She never barks on walks, but sometimes she whimpers when she hears a friend dog bark. Mastiffs are just laid back.
So don't worry, she'll grow into her bark and she'll let you know when people are coming or she's uneasy with someone.
As for favouring you over your husband, it's normal to be warmer to one person in the house and you are it. But, he can even the playing field. He can be the one who is walking her. He can be the one to feed her. He can be the one to play on the floor with her, toss a ball, BUT NOT play tug until her adult teeth have come in, and even then...reasonably. Hand feeding is a bonding technique. Sitting on the floor and giving her one hand full at a time. All of these work. And here's another thought, I bet you are shorter than your husband, and his size may intimidate her. So, getting down to her level may help her see that he is not that scary...he's friend sized.