BTW, 16,000 feet is a little misleading, partly because the signal degradation over distance and partly because the distance between your house and the nearest DSLAM (which may be in their central office, but often is a remote repeater) is not the same as the amount of cable between you and them. The practical maximum is considerably shorter.
Within a mile, you will have at least 80% of your rated speed; no problems.
At ~1.8 miles, you will lose ⅔ of your speed; not good.
At 16,000 feet, you wil have lost over 95% of your speed; a 3MB/second DSL connection will be only twice as fast as 56K dialup at that range.
Unless your local telephone people want to invest a bit of money, there is no chance of DSL there, and odds are that if you live some place like that, cable probably isn’t an option either.
As for satellite… you have to read the Terms of Use VERY carefully and many satellite providers have draconian bandwidth caps and other odd conditions. Before I moved to where DSL was an option, I looked at WildBlue as they seemed to have fewer restrictions and generally not be dicks like Hughesnet.
But it seems that, for whatever reason, your big sticking point is losing AOL. Well, when I went to AOL , it said that you can get AOL access through your broadband connection for $9.99/month. You could have DSL, cable, satellite, or whatever and still keep your AOL.