When MS Project is done I would suggest that:

- you copy all deadlines and milestones into your Calendar;

- at Weekly Review plan follow-ups to meet those deadline and milestones;

- update your Calendar based on daily or weekly meetings with your PMs.

Or you can just rely on your PMs that they control everything themselves and you just meet with them regulary to check if they are follow the due-dates. If not you can change their work priorities for the day or week.