Hi paisley
Perhaps I’m the one that didn’t read carefully because I didn’t realize that you were only looking at innerspring mattresses.
This would be much more difficult to find online in your price range (outside of mainstream brands) although there is a very low cost innerspring mattress sold by Dreamfoam here that may be worth asking them about .
My Green Mattress also sells an innerspring / wool mattress here that has no foam at all and can be ordered either one or two sided that is very good quality/value…
Some of the better options in the New York area are listed in post #2 here and there is a more categorized listing with more detailed descriptions of many of them in post #7 here but you would need to check the websites and do some phone research to make sure any of these had a mattress that fit your criteria.
Phoenix