I think what you're feeling is very common within a few months/the first year of taking on a new dog. What you are feeling is something akin to what many people feel when the y have their first baby and realise how much he or she impacts on their social life etc.
I honestly believe many people feel the same, but don't talk about it - and that can often lead to dogs being handed over to rescue, so well done for being honest and asking for advice.
Keeping dogs is restrictive, that's just one of the things that makes taking one on such a huge commitment and it's something that many people underestimate.
How long have you had her? It takes time to form and build a bond and it doesn't always come naturally. Also, there is absolutely no reason for you to be bearing the brunt of the responsibility. If everyone in the family wanted her and loves her, they should be actively involved in her care. You may well find that she bonds more with another member of the family and that's fine, there's nothing in the rule book that says you have to have the closest bond with her. As long as she's happy, healthy, cared for and has all her needs met you're doing fine.
In terms of finding it hard not being able to go for a day out. Have you thought about finding a local dog-walking or sitting service? If it's just the occasional day out it wouldn't be cost prohibitive and might help you to feel less trapped. Alternatively, do you perhaps have any friends or relatives that may be happy to have her for the day on an occasional basis? I feel for you on this one, because my own dog has separation anxiety and can't be left for longer than half an hour at the moment.
The trapped feeling will be being exacerbated/highlighted by the forthcoming summer holidays, but there are loads of things you can do and places you can go and take the dog with you. In fact it opens up a whole new load of day-out possibilities you may not have considered before - most of which save you money, as it costs very little to head for a gorgeous beauty spot, go for a lovely walk and then have a picnic - especially when you compare it to the cost of going to the zoo or a theme park. Lots of National Trust and English Heritage sites allow dogs on leads. Taking her out to lots of places and rewarding her for being friendly and relaxed will actually help with her not being very forward with other people.
Here are some links for finding out more about dog friendly, family days out:
Dog Friendly Beaches
The Woof Book
The Good Dog Guide
Take Your Pet
The 10 Best Dog Friendly Beaches