, There are some offices that only accept cancellations on the effective day, not in advance. In that case, you might not have many choices. Possibly,
1. As mentioned in the mail, ask your employer to set the due date as 1 April. Apply cancellation on that day before you fly out.
2. Apply for cancellation on 29 March. Immediately apply for a tourism (or whatever reason) extension for a 1,900 baht fee to receive an "application denied, up to 7 days to leave the kingdom" stamp. (Bring your air ticket).
3. Apply for cancellation on 1 April, then leave. (<-this part I'm not sure whether you could have hard time or it is the same procedure as cancellation within the time, or else) You may or may not be fined, but you will get an overstay stamp.
, you can use "fixed account" now as long as it is accessible anytime even with extra fees. You cannot use 'frozen' (<-which is an account you cannot withdraw money from for a certain time) fixed account though.
It depends on your new employer and the local immigration office whether you can apply for a new visa and workpermit inside Thailand. Talk with your new employer about the procedure well in advance.
BUT
You need to make sure you had cancelled your previous extension and work permit officially before you left Thailand.
If you haven't cancelled either of them, you have to cancel them officially first, even if those expired.