At the current time in your budget, I would say that the best NV option for you is digital night vision, which means you will being using an illuminator most nights (digital currently is about as capable as a high end Gen 1 or maybe a low end Gen 2 traditional NV scope).
The best option for you given the money, quality, customer service, etc. would be a Sightmark Wraith 4K and chances are the 4K Mini will work best for you (less native magnification, but plenty of zoom, and a larger FOV). The 4K Max will get you more distance, but the scope is a good bit heavier and has a much smaller FOV, but the performance is a bit better.
The prices on these to vary a bit but assume ~$800. If you go through a small vendor, you can likely score a better price by asking (they can't advertise below MAP by contract, but if you ask verbally, there is a decent chance of saving 5-10%.
The Wraiths come with an illuminator. Depending on conditions, I think it is supposed to be good for 150-200 yards, which is farther than most folks shoot. You can buy a 3rd party illuminator such as a Coyote Cannon and get some additional range and it will cost you a few hundred bucks. Armasight has their new illuminator that they claim is better than the typical flashlight sized illuminators and is nicely priced, but I have not seen one in person.
Here is a video introducing their new illuminator and a comparison. It is now for sale for $220.
The problem with NV and illuminators is that they tend to cause a lot of light to be reflected back to the scope if you have obstructions like tall grass, tree limbs, etc. in your FOV for the light to reflect off of and back to you. Also, when shooting, if conditions are calm, the smoke and debris from shooting will hang in the air and reflect light. In a stiff breeze, you may not notice it. If you have a calm, humid laden night, the cloud can literally hang around (and prevent you from seeing very well) for upwards of 5 seconds.
So if you are hunting coyotes in a big open field, maybe one nicely grazed, you won't have any issues except under calm conditions. If you are hunting a forest or a busy environment, the reflected light is going to be a real challenge. Note, I hunted for this way for 3 years.
This video shows the Wraith 4K MAX my partner was using. We started just before sundown and continued getting video until after legal shooting hours (if we were deer hunting) and the Wraith did well enough that we didn't need an illuminator. Eventually, my partner tried taking a coyote at over 500 yards, using the drop information for the distance from the box of ammo...and he managed to come close. Had he held the proper holdover and corrected a bit for the wind, I think he would have had it.
You said your budget was $2k. You could get out for under $1k, no problem or 1-1.3K depending on what sort of additional illuminator you bought, but I would hold off on buying another until you see if the one that comes with it is sufficient for your needs. It will be for most people most of the time.