Getting a driver position is totally by seniority and the management team has no discretion in that. As far as you passing your 30 days and qualifying, management has a lot of discretion. Everyone makes mistakes, especially when learning new jobs like driving. Your management can overlook many of your errors during your 30 days, or meticulously document each and everyone and use it as a justification for disqualifying you. If your record in your PT operation was truly horrendous, tons of no call no shows, even when at work taking 30 min trips to the bathroom during the shift, hiding out in break rooms or just wandering around the facility at the end of the shift to pad your hours, constantly fighting management if they ask you to perform a task you would rather not, etc, then you will be thought of as basically a turd by your current management team. Rest assured they will communicate that to the Package Center management team who will qualify you or not for the driving position, so that they have a heads up on what type of employee you are likely to be upon qualifying. Right or wrong, this is life and it works this way in most companies and really, any organization of people.
Now, if you just have a few attendance infraction here and there, a small number of service or methods failures, basically just a couple blemishes but nothing truly horrendous, I doubt that would be likely to hurt you.