You must collect your bags in New York and bring them through customs
After customs you can either drop the bags at the bag drop right after customs or you need to take them to a Delta check in counter. Just look at the paper baggage tag that's on the bag. If it says JAX they are checked through to Jacksonville and you can drop them right after customs. If it says JFK (or EWR) you need to take them to a Delta check in counter.
Different PNRs for different airlines on a single booking are not uncommon, but it's probably best to check with the travel agent to make sure it really is a single ticket.