I think unless you plan to spend the whole vacation in Ptown, having the car available is a good idea. There are so many great beaches on the cape, and the National Seashore Park, the lighthouses etc. Not to mention cute towns like Wellfleet.
We just got back, I don't think you will be able to find ANY street parking, or if you do it will be really lucky and not something to count on. There's a reason there are $10 lots all over the place.
Where are you staying that you don't have parking?
The cheapest short term parking lot is the one at the wharf in the middle of town. It's $3 an hour. Of course if you know you will leave the car there sun-up to sun-down, there are lots of $10 lots.