I don't really know the Falmouth area that well, but I can give you some tips if you're willing to drive. That said, even if you do drive somewhere, you should really try to be on the bike as the sun comes up, so leaving your place in Falmouth well before sunrise. By 8 or 9 am, many roads will have constant traffic on them, and drivers on the Cape are generally not friendly to cyclists and unwilling to share the road. If you were going to do a shorter ride in your area, you could maybe get an hour or two in from 5:30 am - 7:30 am on some of the Falmouth roads, but like I said, I don't know them well enough to suggest any.
If you are willing to drive, one potential area is the 'Service Rd' that runs parallel to Route 6 just to the south, between exits 2 and 6. Full disclosure - I have never actually driven this road, but it seems to have minimal traffic and could offer a suitable workout based on the Google street view. The one potential issue here is that you will have some major crossings, and at least one roundabout, as you need to cross main roads that feed off the Rt 6 exits.
If you are willing to drive even further, there are a few good options on a route I typically ride between Dennis and P-town.
Here's a link to a workout I did last year. The beginning on Route 6A is only an option very early in the morning when there is minimal traffic.
For your needs there are two options: 1) Drive out on Rt 6 to exit 12 and park in the shopping center where I turned off 6A. The first couple miles will be a nice warmup as you track my route, then you can start the real work on either Bridge Rd or Herring Pond Rd, being careful to turn on Massasoit when the time is right. This option is nice because there is limited traffic, decent shade, and doesn't rely on main roads. The main drawbacks are the one or two intersections with stop signs and sometimes bumpy roads.
2) Drive out on Rt 6 to around the Wellfleet drive-in movie theater, which is near the end of Option 1 above, and park somewhere (should be plenty of decent options). From there, Rt 6 has a decent shoulder and you should be able to bike for miles and miles, pretty much all the way out to P-town. The drawback is that riding on Rt 6 absolutely sucks. It is safe, with a wide shoulder, but it is boring and has lots of traffic (and some traffic lights, depending on which section you ride). It is also generally not flat, so you'll have long gradual uphills followed by long-ish downhills. Not a problem as far as getting your workout done, but you'll have to have some decent speed to keep your power up on the downhills. On my ride, I divert off 6A a couple times, but just because I like to minimize my time on there.
I realize none of these options are great when you're starting out from Falmouth, but this weekend (or really any weekend Memorial Day - Labor Day) will be tough to find good/safe cycling.
Edit: spelling