Which market in New York, New York City?
Unless you are in the New York City DMA back home, you are most likely not going to be able to get your HD local channels in Florida. The HD spot beams are way smaller than that. If you do live in the New York City DMA, then your locals are carried as CONUS beams instead of spot beams and you should be able to get them (they wouldn't be coming off the 99s or 103s transponders).
The SD local channels come off different satellites and have different spot beam maps that I think are much less localized, that may be why you can still get your SD local channels there.
If you have a SWM dish, or SWM multiswitch then you should only have one coax cabel hooked up to any receiver or DVR, so that shouldn't be an issue (unless you don't really have a SWM dish/multiswitch).
You said this is your second home, are you going to be there for a while? If so call DirecTV and have them move your service address to your vacation homes address, then you can get the local channels from there. This is the proper way to do this. DirecTV won't have an issue with this assuming you only do it a few times a year. If you are calling every weekend or so to change the address then they will likely make you get a second account.
All this said, it is against DirecTVs Terms Of Service, to have accounts active in two places at once. So if you are at your vacation home watching DirecTV, nobody should be watching it at your primary home back in New York. Doing so will get your account suspended/cancelled if they catch you.