My $0.02 worth...
Could someone recommend a reputable closing company with reasonable fees.
Although I have no affiliation with them (beyond that of being a satisfied customer in multiple transactions over a number of years), I happen to like Timeshare Transfer, Inc. of Vero Beach, FL (...can I say that without appearing to advertise for them?). They don't handle all states, but you can look at their web site on your own to easily determine whether they handle whichever state it is in which your particular timeshare is located.
Also, I have no personal experience with them, but many folks speak highly of JRA Closing Services too.
You should probably expect to pay around $300 no matter who you choose to utilize (excluding resort transfer fees, which vary considerably). Title searches and title insurance (if you care to have either) will drive the cost upward from that approximate starting point.
I'll refrain from bad mouthing one other closing service which, although competent, is painfully slow. They are located in Bozeman, Montana. ;-)