Hi drnancymiller,
The first place I would start with any new mattress purchase is post #1 here. This and the information it links to is the most important single post on the forum.
One of the links in it goes to the mattress shopping guidelines here which will help you aoid most of the worst choices of both mattress manufacturers and retailers including all the stores and brands you have mentioned.
Once you’ve read some of the information in this post … you will be much better able to shop based on the quality of the materials and components in a mattress instead of by brand (which is only the company that puts the components and materials inside a cover). you will also be much more able to recognize the “experts” and salespeople who will educate you about their mattresses and what is in them instead of just “selling” you based on marketing stories with no real substance. These are the people who already know what you would otherwise need to learn to make good choices.
Post #2 here has a list of the better options and possibilities in the NYC area and post #7 here is a more “categorized” list with more detailed descriptions of many of them.
If you are in the Setauket area then post #4 here may also be helpful although I would tend to first call and then visit some of the manufacturers or retailers in NYC anyway.
Even the bottom part of your budget is more than enough for a very high quality/value mattress that uses very durable materials.
Phoenix