I doubt anyone will ever check your age, unless you try to buy a car or an "adult beverage."
The advice for a prepaid credit card or a debit card in your own name is good advice. Leave your Dad's card at home. Less hassle all around.
I gather you are tenting, so your don't need full hookups. I'd look for state, county, city or national parks for cheap campgrounds. Corps of Engineer campgrounds tend to be great, too. You might also consider a Federal interagency annual pass. My recollection is that they are like $80 a year, but get you free admission and half-price camping fees at COE and national parks. Think the whole summer for $8 a night. You are too young for the Senior Lifetime pass ($10 at age 62 -- best $10 I ever spent!), but still a great bargain.
If you have a smart phone, get one or more of the camping apps for it to help you find campgrounds. The Allstays app costs a few bucks, but has more locations with less info. There are other free ones, including one for federal facilities that links directly to the reservation system and can check for open sites. These apps are for both Android and iOS devices.