Ebay seems to be the place where you get the most viewers and competition. The walk-in coin-shops only act as a middle-man between you and the end buyer/collector. So there's going to be their "cut", that is eliminated if you go with ebay. Because then you're selling to the end-buyer. Or ... at least .... a dealer who is willing to compete with that mind-set anyhow !
Far greater chance of getting scammed off ebay though, I'd take a reduced cut at a walk in shop rather than throwing it in the lap of the gods on ebay.
Yes, I know there are successful transaction there all the time, I used to be a power seller. People get burned too, and ebay almost always tends to side with buyers in disputes.
Professionally graded Mint State 60 to 61 1890 S Double Eagle should bring at retail about $1500 to $1800. Maybe more with the recent rise in gold. Selling your ungraded example, you are likely to get 15% to 25% less. If you sell to a typical pawn shop, up to 70% less. Ask around. Try to avoid FeeBay except to look at completed auctions to see what the average selling price has recently been. I prefer coin clubs, if possible, for ungraded gold coins.
go to the library and look up the World Coins book. In the front and back are ads from coin buyers/dealers. If you need an email address I have 2 or 3.
I would use a reputable coin dealer. Ebay is ok, but to many variables could go wrong with a coin of that value. It may even be lost in shipping. Just because it has a tracking number does not mean they will be able to find it if it is lost.
Just as an aside, most reasonable folks aren't going to buy a rare gold piece from just anybody. If you really want to sell an authentic coin like that, go through a legit auction house or dealer.