The Hunter 45cc (which you mentioned below) has either a 5' shoal or a 6.5' draft. In the case of the ICW (and St. Augustine) less is better. The vertical clearance of 61'-4" for the rig (assuming furling) needs to be adjusted in your mind to about 63' (or more) for instruments and antennas at the masthead.

http://www.torresen.com/sailboats/hunter/45cc
http://torresen.com/s4s/brochures/hunter/45cc.pdf

If you leave earlier in the later Spring, you more chance of winds out of northwest rathen than south or southwest winds. I think you're likely to be facing oscillating winds as the fronts move through every three to five days. (ALL YOU WEATHER GURU's WEIGH IN HERE.)

Scenario 1 - Delivery Captain (grease this pig and get it out of here)

In this case, you'll go offshore along the coast to at least Norfolk. Then, unless weather should force you in, you'll take a shot around Cape Hatteras (the Mean Joe Green of capes on the East Coast) and Cape Fear (didn't get it's name for no reason). Thereafter, you have some spaced choices to duck into -- but enough room between yourselves and the Gulf Stream to continue south outside. Alternatively, you'll take the ICW until below the Capes and then go outside the rest of the way.

Scenario 2 - More relaxed approach (as Dan Y and others might say)

You head down LI to the East River, time your flow through Hell Gate so you get "flushed" out. You can stop at the Sandy Hook, NJ (just beyond the Verzanno Bridge) Narrows if you need to; or continue South to Atlantic City or Cape May. Then Up the Delaware to the C&D cannal. Down the Chesapeake Bay to Norfolk.

Then ICW to below the Capes. Then you make your own choice -- a little outside or little sight seeing.

There's a nice ICW Waterway Guide( or Guides) that you can get that can give you some guidance.

DON'T PLAN ON MOTORING IN THE ICW IN THE DARK!

Why don't you tell us a little more about the crew you'll have with you and the equipment on the boat?
You can plan on leaving the boat and breaking up the trip down -- so that you return and continue your adventure.