Absolutely agree -- if you want to be competitive, you have to be well mounted.
That being said, if the OP already knows how to move and handle cows, then she needs to be mounted well enough to get there.
The way we put a 'handle' on a horse is to teach it from day 1 to move its shoulders. If that takes a short bat or tap on its shoulders or a spur at the girth or ???, we strive to get a horse 'light' on the bit, responsive to leg and rein, and to MOVE its shoulders. Did I say it has to move its shoulders? It is all in shoulder control.
The best tool for teaching this is one critter (can be cow, steer, heifer, bison -- my favorite or even a goat or llama) in a 150 foot round pen. You teach the horse to drive the critter forward, then walk parallel to it and finally step a step in front of its 'drive line' to stop it. When the critter stops, you teach the horse to stop with it. When the critter turns and goes the other direction you teach the horse to back up one or two steps straight (this 'loads' its hind end) and you teach it to roll back toward the critter. This should be a 'true roll-back where the horse steps BEHIND its front foot with the other front foot. [Most reining roll-backs are not true roll-backs -- they are just 1/2 of a turn-around or spin.] It should learn to 'sweep' around in front by moving its shoulder the full 180 degrees or close to it. Then, when the horse is facing the direction the critter is facing, the rider drives it forward to catch up. If you have a horse with any 'cow' in it, it will start hurrying the roll-back on its own.. It is best if the horse 'wants' to get turned around and you only hurry it to catch up. They stay more correct that way.
The big thing you want to teach while the horse is going parallel to the critter is to keep its head looking slightly toward the critter and keeping its ribs bent out away from the critter. In other words, the horse is moving with reverse bend, coming off of the rider's cow-side leg.
The other big thing is to NOT LET the horse flop its butt out (away from the cow) when the cow stops. The horse MUST learn to hold its ground behind, keep its ribs and shoulders out (away from cow), stop while staying parallel to the cow, load its hind end by stepping back, sweep the ground to change directions by only moving its shoulders and then catch up to the cow. When you go to catch up to the cow, the round pen lets you make a smaller circle so you do not just 'chase' the cow. Your cow-side leg should push your horse away from the cow to make that circle smaller while staying parallel to your critter.
You do not teach this, even to a talented horse, overnight. If you take the time and patience to teach a horse to stay correct, you will soon see how much 'cow' your horse has. The ones with a lot of cow try to do too much too quickly. The disinterested ones will just not learn to stop when the cow stops.
If you can afford it, the best way to start sorting is to buy a horse that is already pretty well trained. It is a lot easier to learn where to be when the horse tries to get there for you.
As for finding someone to practice with ---
NY is not exactly like living in OK or TX. Are there any Ranch Sorting or Team Penning competitions near you. If you can find a competition near you, I would just go and get acquainted. Find the people from your area and ask if you can come and play with them. Most people will let others practice and play with them if they pay a 'stock charge' to off-set cattle costs and feed.
We love our bison. I have had two or more here for about 15 years. I have two now. One Bison in a round pen will let you teach many horses how to be where you want them to be. A 150 foot round pen can even be used with two riders and two horses. One rider just 'works' the other horse and rider the same as if it was a critter. This is obviously not as good as working a cow, but you can give a horse a reason to be where you want him to be. A 'flag' does the same thing. Horses learn to work them and they give the rider the chance teach the straight stop, the step back and the roll-back.
Hope this helps.