Unresponsive advisor, job offer in limbo
I'm going to just guess that meeting your tight deadlines will be impossible with an advisor on sabbatical. So you need some short term action that will preserve your position. 

You can certainly ask a new employer about starting before you have officially received the degree and they might be fine with that, or not. But it would be more likely to be a positive decision if they had some quasi official assurance that your timely and successful completion is not in doubt. 

You might try one or both of the following. Ask your advisor for a letter, assuring the reader (your potential employer) that completion is a pretty much pro forma situation. "We need to dot some i's and cross some t's etc." 

Go to the department head and get confirmation of the above and even his/her help in getting that letter from your advisor. The head might also include an additional supporting letter, confirming that your progress is on track, etc. 

Many employers would be supportive as long as the situation doesn't seem like it will go on and you wind up ABD or unable to do the job you are being hired for. Of course, there may be some situations in which there are firm, legal, requirements that you get the degree before starting the job, but those are likely rare. But I'm assuming this is an industrial job, not an academic position, which might well have firm requirements to start. 

Another option, perhaps, is to ask the employer for a delayed start date. But that request would also be positively supported by the sort of letter(s) mentioned above.