Log cabin hotels

What port is your ship arriving at? Please know that the US public transportation is much, much more limited than in the UK. It may be very difficult to get to a log cabin themed hotel by train. You will probably need a combination of train and local buses/taxi. Is renting a car an option? Do you want to stay in one place the whole time?

What do you consider reasonable rates? Keep in mind that you are traveling during the Christmas/New Year’s holidays and the places you mention are big ski resort areas that will be in peak season.

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Travel4fun, unable to see your reply, have clicked on your moniker, but nothing

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I googled "log cabin hotels near new york city" This was a result.
https://glampinghub.com/rentalsearch/winter-log-cabin-rentals-upstate-new-york/
I think the first step that you need to take is to determine which are the cities/towns that are accessible directly by train from New York City. Here is an interactive Amtrak planning map.
https://www.amtrak.com/plan-your-trip.html

Then do an AirBnb search or a booking. com search for those locations.

Perhaps someone who lives in the Northeastern US will know of such a hotel, but most of us would have to search using the internet and I think you might try that approach.

I am from Maine and have traveled all over New England. I can confirm that it will be next to impossible to explore those states via train and/or bus. Via Amtrak you can get to Boston, Burlington and Portland (changing trains in Boston by traveling cross town from South Station to North Station). Travel by bus won’t offer many additional options. You will need a car to see anything outside of city centers particularly in Burlington and Portland. You should also know that many sights outside of the larger cities close after the summer/fall tourist season. Several shops and restaurants in popular coastal towns may also close after the tourist season. Double check your destinations so that you won’t be disappointed.