First the box spring needs to be replaced. There are real reasons for it because they are not made the way they were 10 years ago. You don't need to spend $2000 for a great spring mattress, but some brands are going to be more than others, so be open about the brands. I'm very partial to Laura Ashley mattresses, but I don't know if you can get them in NYC. I also like Tempur-Pedic a lot, but that will run more than $2000. If you post this question at myfurnitureforum(dot)com, you'll get more detailed answers from some picky retailers there.

Royal-Pedic is the company name and it manufactures in Cal.
We have the 8" all cotton/coil mattrees and box springs with a 4" latex/wool topper and it is the best purchase we have ever made. I have had severe back problems for many years and this product has given me great relief and wonderful, restfull nights. My wife has also indicated that this set sleeps much cooler that our previous S&F pillow top set. Find a retailer and try out this brand before you make your decision (we also looked at the brands you mentioned along with others including Hypnos).

Also try the link posted in the previous reply, Duane is a retailer who carries Royal-Pedic and has significant knowledge and insight.

Yes. Replace the box spring. You may void the mattress warranty if you don't...and the box is the cheapest part...