There is very probably nothing wrong with your trailer, so don't get upset with the seller just yet. There are a few things I would like to add/emphasize to/about the info in this topic (I have been towing all sorts of things with all sorts of vehicles for many years and try to be very familiar with what can go wrong):
1) Fishtailing does not come from towing capacity of your vehicle, but rather wheelbase/tongue load/tongue length. A short, light vehicle towing a longer, heavier trailer is a recipe for problems, and as several folks have said here, you need the proper tongue weight on the hitch. If your hitch was "popping" ("bang" sound and jerk) when you start off from a stop, your tongue weight is probably too light (hitch popping up off the ball and jerking back onto it when you move). You will also have issues if the distance from the end of the trailer tongue to the trailer axles is too short. This can also cause fishtailing.
a) Just FYI--towing over the capacity of your truck will wear out your transmission and engine quickly, but the main issue you will see is with braking. Most people don't think a load is any problem if their truck will *pull* it, but just try to make a quick or panic stop with that load pushing you. Plus, you will wear out your brakes quickly and warp rotors. YOU ARE A DANGER TO YOURSELF AND OTHERS WHEN YOU TOW OVER CAPACITY OF YOUR TOW VEHICLE.
2) You need to make sure your whole rig rides level with the full load in the trailer and truck. Many people discount all the stuff they throw into the back of the truck, but that all adds to your rolling load. Your truck should not squat or bottom out with the trailer loaded properly. If it does, and your truck is rated for the load, the best thing to do is add a set of air-bag helper springs to the back of your truck. They can have air added for trailering and removed for regular driving. The trailer also needs to ride level for safety, tire wear and the comfort of the animals inside. Be sure you have the right drop on your draw bar (part with the ball on it that goes into the receiver on your truck) to make the trailer ride level.
3) Trailer brakes are also essential, especially when towing a heavier load. I would personally never put my horses in a trailer with no brakes. You must have a brake controller mounted in your vehicle and you need to make sure it is set properly. Used correctly, trailer brakes can help correct a fishtail caused by other means, but are NOT the answer to the fishtailing problem.
4) You MUST also have the proper rated tow equipment on your truck. It needs to be at least Class 3 (rated for 5,000 gross trailer weight and 500 pound tongue weight) and maybe better, depending on your load weight.
5) You also MUST have and use safety chains properly. If the trailer hitch breaks away for any reason, the chains will keep it attached to the truck and not running off across lanes of traffic. What usually happens is the hitch breaks away, the chains catch the trailer and jerk it hard, you respond to the jerk by slamming on the brakes, and the trailer rams into the back of your truck and both pieces come to a halt. Otherwise, the hitch breaks away and the trailer falls behind the truck very quickly, often dropping the hitch into the pavement and flipping the trailer, or it takes off into traffic and kills people.
6) Anti-sway bars and load-leveling hitches are very helpful when towing larger loads, but all of the above must be done before you can consider adding them. They do not make up for improperly loaded trailers or inadequate towing equipment, but can provide safety and comfort when things are done correctly.
I hope this helps. Trailering, especially with live animals, is not something to be flippant about. You MUST use the right equipment, and properly, or you WILL end up having a very nasty mess at some point--there's no two ways about it. I would strongly suggest going to someone who knows about this stuff and can help you (not someone selling trailers and equipment who wants to get $1,100 out of you for anti-sway equipment).