Is there a reason you can't have your ceremony at the Seaport? They have a non-denominational church. I don't think they do any services aside from the weddings, though I could be wrong.
Ditto PP... We had our reception in the River Room/ North Lawn at the Seaport. Our ceremony was off site because we wanted a Catholic Church but, originally, we had the non-denominational Church booked on site. Like PP said, you can also have it on the North Lawn if the weather is nice (I think they'll let you book the church as a backup).
We also looked into using the chapel at the Seaport, although be aware that it is not all that big so you'd need to figure out if it works with your numbers. There is also a small church in Mystic Village, that could be another option for you.
Aside from those choices, if you are not tied to a particular church I would not recommend trying to use a local church that holds active services, they will either charge you a lot (because you are not a member) or they may not allow it, and they will likely not want you to bring in your own officiant - basically, if you don't have a solid affiliation the local churches will probably make things difficult for you, and if you aren't attending church regularly anyway then it may make more sense for you to get married somewhere that is not a church (unless it is one of the two mentioned above, which are 'neutral spaces' of sorts).
Good luck with your planning, hope you find what you want!
Re: Mystic - Ceremony Sites
June 2012 Bride!
Love love love the Seaport, btw!!!
Aside from those choices, if you are not tied to a particular church I would not recommend trying to use a local church that holds active services, they will either charge you a lot (because you are not a member) or they may not allow it, and they will likely not want you to bring in your own officiant - basically, if you don't have a solid affiliation the local churches will probably make things difficult for you, and if you aren't attending church regularly anyway then it may make more sense for you to get married somewhere that is not a church (unless it is one of the two mentioned above, which are 'neutral spaces' of sorts).
Good luck with your planning, hope you find what you want!