If you have just spent a few hours in the boat then I encourage you to stick with it. You can learn a lot about paddling and yourself (frustration level) trying to make a boat go straight that seems to have a mind of its own.
A good boat body connection will make the boat more responsive to your movements. Getting an unruly kayak to go straight is also about lower body movements within the boat, not just paddle strokes. Rockered boats naturally want to spin out as soon as you stop paddling. A good experiment is to get the boat up to speed with five or six quick strokes then stop paddling and count the number of complete spins/revolutions the boat makes. Be prepared for a feeling of instability as well. A whitewater kayak will often spin completely around more than two times. In fact I often tell newbies that ww kayaking is just about learning to control the spin of the boat. You might try shifting your weight a bit off center to help counteract the veer once it starts.
It is also really important to keep your strokes short (not extending past the hip) to go straight. You can stroke past the hip to use correction strokes (stern rudder strokes) but realize the goal is to try to make the boat go straight without having to constantly rudder, pry, or draw at the very back of the boat.
I’ll be the first to admit that if you put me in a long skinny boat that I’m going to struggle a bit with the lower initial stability and “pinned” ends that don’t want to turn. So different boats really behave quite differently. I just haven’t put the time in with that type of boat but conversely after paddling a taureau canoe (very short, very highly rockered, single bladed paddle) for a couple of years, I think I can make just about anything go straight. So even with a slightly deformed hull, and rockered ends you should still be able counteract the veer in your kayak without a skeg. It will be challenging. In fact I find paddlesports in general to be challenging. Here is a whole list of things to try and experiment with. You might just want to focus on one thing at a time.
First, ask yourself this, Is the boat traveling faster because you are doing more strokes (at a quicker rate) or by doing bigger (longer) strokes? Speed is going to expose/accentuate more flaws in your forward stroke. Many folks lengthen their strokes unintentionally when trying to get a boat up to speed. So focus on doing more short quick strokes from the toes to hip. Strokes should be parallel to the boats midline not the sides of the boat. Realize that when you place the paddle blade toward the end of the boat, where the boat is unpinned (rockered), that it will naturally turn. So If you want to go faster just think more strokes not powerful (longer) strokes. Also when you are pushing your top hand out make sure it doesn’t extend across the midline of the boat. Think about picking grapes with your top hand extending out in front of your chin, not to your side.This may help keep the paddle blade more vertical (not sweeping out and pushing the boat away from the side you are paddling on). If you’re worried about shoulder rotation, try paddling a bit more stiff armed for practice. It is actually pretty difficult to get good rotation and not extend the hands beyond the midline of the boat.
So in summary: 1)short strokes, toe to hip, no further unless correcting 2)more veritcal strokes, parallel to boat’s midline not sides. Horizontal strokes are turning strokes so more vertical strokes are straighter3) If the boat veers experiment with shifting your weight slightly to counteract the veer. Let us know how it goes. I never discard (trade/sell) a boat until I’ve paddled it at least 20x. Each boat can be a new learning experience. No boat does everything well- your current boat likes to turn instead of go straight …that doesn’t mean it’s not a good boat, it is just a bigger challenge (to go straight) than some other boats and you’ll be a better paddler because of it. New paddlers blame the boat, old paddlers blame themselves. All of my fleet as deformities, oil canning, I’ve even gotten a broken skeg on my crossover. Yet they all still work. Hang in there, try a bit more…give yourself permission to struggle a bit. 40 + years later I’m still enjoying the paddling “struggle”. Since my hip replacements I’ve practiced rolling well over 60 sessions in the past two years, and quite frankly still struggling to get up. Maybe after another 20 sessions I’ll consider an easier boat (which is what others suggest)…but then I wouldn’t learn nearly as much. Some boats you grow out of. Your boat is one you’ll have to grow into. keep going, you haven’t yet reached the “suffering needlessly” threshhold.