The deeded perpetual time share is in North Conway, New Hampshire, at the height of fall foliage season, Columbus Day week in October. The Resort will not help us to sell it as a resale, so we will need a reliable firm to do it for us. We do not want to be waiting for years. We hate to give it up, but we just cannot get up there every year as we once did.

Timeshares DO NOT sale quickly, it just doesn’t happen. You can not even give them away quickly.
There are a lot of scams out there offering to sell it quick. BE CAREFUL.
To resale or rent a timeshare is a hard task, the basic rule is not to pay money up-front, don’t be another victim of a scam. There are some free classified adds where you can post it and get some results:
http://timeshareownerscommunity.com/
http://tug2.net/
http://craigslist.com
http://ebay.com
http://recycler.com/
http://usfreeads.com/
http://www.inetgiant.com/
Sell it on eBay — auction it off. There are a number of companies that sell timeshares on eBay.
Advertise on craigslist.com
If it is part of RCI or other exchange organization, spacebank or trade it for a week elsewhere. Or rent it out the years you do not use it.