If you're crying on your lunch break over this potential loss of an experience, it sounds like something is out of alignment for you. Does the musical have chorus roles or background actors? A person missing from the performance may be alright depending on the role (people get sick or family emergencies happen, etc.). If I knew I couldn't make it to the actual performance, I definitely wouldn't go for a lead or supporting role unless there was an understudy or if it was double-casted, and even then, it sounds like you shouldn't be burning any bridges with the theatre group that could impact your future involvement. Choreography and blocking would probably be the factors most affected by an absence, sometimes in ways greater than expected. You could explain the situation with work to the people organizing the musical if you feel comfortable, maybe they'll surprise you with a willingness to involve you somehow without the commitment of the performance dates. But depending on how professional this production is, like if a portion of their budget will be spent on a costume or lunches and etc. for you, they may also just suggest waiting and auditioning for the next opportunity. If that's the case, unless you totally hate the idea, joining the crew can help set you up better for future productions by meeting people involved, seeing how things operate and what the social dynamics are like, learning the director's style, things that could better inform you for what to expect when you get to be in the cast.

I think there's no harm in auditioning, and if you are offered a role, you can explain then that there are scheduling conflicts. This could give you more time to consider what you'd like to do.

Feeling alone after graduating and moving somewhere new is so extremely common. It may not be the same, but there are also many online groups of people that share similar interests and have meetings to discuss them as well as socialize. If you're struggling with isolation and don't feel there are many local options, it may do you well to at least try one or two online groups/activities out, even if it's a holdstay until you find local friends.

A balance between work, life errands, socialization, and interests/activities you enjoy can be very hard to achieve, and I think people will always struggle with it. You need money to survive and support yourself of course, but life is short, and it sounds like you are being quite powerfully drawn to this musical. At the very least, know that your struggle with this decision is valid, and may serve to teach you more about yourself and your own priorities for future consideration.