I have seen this posted all the time. Put in my two weeks and company gets upset that your hurting the company.
First always keep in mind, employment is at-will, you can just leave without notice. You can always leave sooner than your notice if it's causing you stress.
Second, if your that important they need to pay you more… Use this as an opportunity to open negotiations, just because you have accepted an offer with a new company doesn't mean you cannot back out. If they are unwilling to negotiate then your not that important and your not responsible for the company…
Third, if the company really really needs you to stay. Then offer entering into a short term contractor position. This actually happens a lot… It's not unheard of or crazy. You should expect them to pay you substantially more than you currently make (keep in mind you will not have health care so factor that into the pay). You can likely kick back your hire date with the new company and get a nice big paycheck on the way. It also really does help your former company and keep good relations.