I have always left a company by scheduling a gap between the old employer and the new one. I have a pension coming in, so can afford to leave on, say, January 20, and start the new job on March 1, or something like that.
But I think a better way would be the following:
Get a new job without telling old employer, and for the orientation period, take your vacation and sick time. You will get paid from your old employer while working for the new one (and making sure the new one is worthy of your efforts).
When your vacation is over at the old job, go in and tell them you were offered an opportunity, but it requires you to begin right away. You're very sorry, but need to pack up your stuff, and can help train-up someone on your stuff that day, ad will be available over the phone or email. Truth is, your job should have had a backup doing it while you were on vacation already, right?
If your old boss is unhappy like you're stealing from him, then you know they suck balls and deserve to be left behind.
If your old boss wants to beat the offer from the new company, hear it out.
Do not feel bad whatever you do. This is the free market at work.