I am a huge fan of SxS doubles and have shot them since I began hunting (some 50 years ago). I believe you will be best served in the used gun market.
My first choices would be the L.C.Smith or A.H.Fox guns, but I am a real admirer of the Spanish doubles. AYA is probably the best known and top quality, but many of the less expensive and lesser known guns are a great value as well (and can be had for a song). The down-side to the Spanish guns (for me at least) is that they typically come with an exaggerated "beavertail" forearm. I much prefer the traditional "slinter" forearms or at least a milder form of the beavertail (I know they call it a "beavertail", but does it have to look like it was just cut off the beaver).
This may come as a surprise, but I really like the "brand name" guns made by Crescent Arms. Best known is the American Arms Knickerbocker, but you will find them with lots of different brand names. These were very well made, but inexpensive guns when originally made. The "trick" is that most have been used very hard. If you can find the rare, less used Crescent....you will have a true prize.....for very little money.
My own upland game gun is a super clean Crescent in 16 gauge. Nice checkering, sidelock action and light weight. In fact, for your intended purpose the 16 gauge might be something you should consider. Ammo is harder to find and often a little pricey, but the cost of 16 gauges is likely much less than a similar 12 gauge.
