Howdy
If you are planning to shoot Single Action Shooting Society (the largest cowboy shooting organization) matches, the only pump shotgun that is allowed is the Winchester Model 1897 and its replicas. No other pumps are allowed. The Chinese made Norinco and Interarms replicas of the Model 1897 are also legal. The Winchester Model 12 is legal for Wild Bunch shooting, but that is a specialized version of CAS and it is not legal for normal cowboy shooting. With all due respect, shooters who are proficient with the '97 do not load two in the magazine. Instead they load it as a single shot from the start, throwing new rounds into the ejection port as they work the slide to eject an empty. This requires a lot of practice, but it is the way all serious competitors use a '97. Guys who are good with a '97 can knock down a tombstone target with one shot and reload for the popper while it is in the air and still hit the popper. I have seen it done many times. No, it is not in my skill set.
The '97 does have its own unique set of problems. The originals are all old guns, the last ones were made in 1943. Mine was made in 1908. So any original you buy is likely to have a lot of wear on it. There are over 100 parts in a Model '97, and Cowboy Action Shooters tend to treat them very roughly, so the '97 can break down and leave you high and dry for the match.
The Winchester Model 1887 lever action shotgun and its replicas (the gun Arnold was shooting from his motorcycle in Teminator II) is also legal, but they tend to be difficult to use and I do not advise a beginner using one.
Given your budget of $600 probably your best bet is a Side X Side made by Stoeger. For some reason, the coach guns are very popular, probably because of the myth that coach guns were popular in the Old West. Personally, I would choose a shotgun with a longer barrel as it makes it easier to hit the targets, particularly the occasional flyer. If I was in your shoes I would probably go with one of the Stoeger Uplander models in 12 gauge with either 26" or 28" barrels. I would probably stay away from the one with the black stock too, just so you didn't have to explain over and over again that it is wood. A word of warning, Stoegers tend to be pretty stiff right out of the box.
http://www.stoegerindustries.com/side-by-side-shotguns
Just so you know, to be SASS legal a Side X Side shotgun must not have automatic ejectors. Extractors only, no ejectors. Single triggers or double triggers, both are legal. Must be at least 20 gauge and no larger than 10 gauge. I recommend 12 gauge. External hammers or internal hammers, both are allowed. My favorite CAS shotgun is an old Stevens hammer gun made around 1906 or so.