If Ocean is snotty you can limit your run to Fire Island Inlet and shinnecock then come inside. Shinnecock Bay and Peconic are great but then you deal with canal.
If you have never done the canal, you should know a few things, pretty basic stuff. Canal is easy if open and slack. If closed, then pay attention to lights and which side you want to fender on, don't tie off, just loop rope and hold. If Open and running like a rapid river, then power up to get thru and hang on tight! If the opposite direction, there is a chance they will make you use the smaller canal, running North to South, it is very tight, again, just get boat straight and go thru the speed you need to stay straight.
Fire Island inlet i hear can get low at low tide these days. Favor red side of buoys, follow channel, try to catch high tide.
Shinnecock inlet can get very hairy. I have seen huge swells in there, again, pick your path, and be prepared to take some waves till out or in depending on path.
As for your fuel question, by Montauk, go to Star Island, it is easy fuel stop. There is another one that is inside more to the right when coming in. Montauk yacht club I am not sure they have fuel or if they do I am sure it is a rip. All fishing boats go to Star.