Well, if you've already handed in your resignation at your current employer then I don't think you have much choice.
If you want to do the transition then you need to do it with three other people in the room: the business project sponsor, the PM's manager and the PM. If any of these people are unwilling to be present for the entire review and and handover process then you have a serious issue on your hands. They should be treating this as a critical event in the project and their absence is an indication that the project is not important.
For the review, the current PM should bring the latest version of the schedule *and* the previous version. Insist that the previous version be at least two weeks old ... otherwise you're likely to see an update and then the update of the update with only trivial changes.
You must also insist on reviewing the the current issues list.
You need the two versions of the schedule to get a sense of how the project size is developing over the period of time between the two versions. You also want to get a sense of how frequently the schedule is being updated ... I'm guessing that it was probably done in a rush recently and hadn't been updated for some time.
If the schedule is seriously out of date (and you can only determine that with some serious questioning of the current PM) then I'd start to get cold feet. You need to make certain that existing dates are valid, that work efforts and task durations are reasonable and that the resources are still working on your project and haven't been transferred elsewhere.
With the issues list, you need to know what issues haven't been added to the list ... there are always some. You need to find out the status of existing issues and you have to find out which of these issues are (or will soon become) show stoppers.
Natually, under normal circumstances these two items are just part of a much larger handover process.
One thing is sure: if you join this company, your working life won't be dull.