We have enjoyed crowdless camping along River Road many times this time of year. Should be a great time...let us know
Of course, as you mentioned it is getting more and more crowded all the time.
Reserve America is a joke and you are correct on phoning in. I usually wait till the day before and chance it with the park staff. Very seldom get let down and usually end up better! They can pull a few strings and accomodate most times.