My first shotgun was a youth model Stoeger 20 gauge SxS. At first, I couldn't hit the broad side of a barn with it until suddenly, everything clicked. I used it for years after that a rarely missed quail and dove with it. You can't shoot O/U and SxS the same way. I think the SxS has declined partly because people naturally shoot O/Us better. One theory is that most American's start off as riflemen and O/Us fit more like rifles so we are used to that feel more, even if you aren't aiming as you would a rifle. I realize this following may be controversial.
Dangerous game hunting with a vintage double rifle is what brought me back to the SxS shotgun. Realistically, I don't think SxS guns have many advantages over an O/U. The first point below is usually true in general. The others are as well, but that depends on the proficiency of the shooter.
1. They are typically lighter than O/Us.
2. SxS actions break open at a shallower angle than O/Us, theoretically allowing for faster reloading.
3. SxS handle like an exotic sports car. They are smooth, fast, and responsive to the shooter when the shooter moves the barrels first, rather than the stock. As with dangerous game, moving the barrels towards the target BEFORE moving the stock to shoulder results in a natural and smooth swing that puts you in target faster than setting the stock in the shoulder and moving the gun to the target. Stock dimensions play are part here as well, but there are very few deep pistol grip SxS shotguns compared to O/Us. The English straight grip probably does best to force the shooter to move the barrels first due to lack of grip purchase on the stock.
4. They balance better. Though infrequent, SxS guns that are heavier than a O/U may still feel lighter due to the balancing point (center of gravity). A heavy SxS will still feel lively and responsive in the shooter's hands.
5. I have heard and read many articles where people say the advantage of the O/U is a single sighting plane. I think that's hogwash. I have never seen a firearm with anything but a single sighting plane. You don't sight a shotgun anyway so it shouldn't matter even if there were two sighting planes. The O/U stacked barrels allow for more peripheral vision but that is sometimes good and sometimes bad. You don't want you eyes to be distracted from the target. I think it's a wash between the two. Once you get used to seeing a target with a SxS and having less in your periphery, it doesn't affect your ability to track.
6. The are romantic. They ooze nostalgia and the lines of a SxS are simply beautiful. You may shoot an O/U better, but the SxS will give you more emotional satisfaction.
As far as budget friendly guns go, the answer is always "it depends." If your budget allows for a $10,000+ gun, go for it. If it's closer to $200, I'd save my money some more. If it's somewhere in between, you can pick up some great Lefever, Steven's, Cogswell & Harrison, L.C. Smith, Beretta, and other names for $2,500-$6,000 depending on gauge and condition.