you can't have people eternally be on waivers, and I don't actually think that is what you are looking to do as people should be able to pick up players that nobody wants without sacrificing their waiver priority. Yahoo has an option to place all free agents on waivers when the first games start on sunday and then they come off waivers on Tuesday - this eliminates people from picking up players mid game (except for when there are thursday games etc). Unfortunately, this feature is only available in the yahoo plus leagues and not in the free leagues.
If it is a major issue and you don't want to upgrade, consider limiting the player aquisitions to a really low number, so if people jump on people week 1, then they can screw themselves later on in the year if they have an injury and can't pickup a quarterback (for example). Or upgrade the league or find another site to use next year - I know that fleaflicker does the sunday-tuesday for free.