I've taken a counter-offer before. It worked out perfectly fine, contrary to common advice. My raise was equal in magnitude to yours.
However, the company was not in any sort of financial trouble, there was no shuffling of management, and I had a really, really great relationship with my manager, who just really did not want to see me go. My reason for leaving was also 90% financial, 10% other stuff which wasn't all that big of a deal.
People told me still not to take it because "if you have to get another offer to get a raise, is that a company you want to work for anyway?" My response to that was: (1) I always got great yearly raises, just not of the 50%+ magnitude. (2) I didn't even know my market rate was as high as it was, so I didn't know to ask for it; (3) it's just business, and there's no sense in being overly offended, especially if you're being treated well to begin with. You can't blame a company for not doubling your salary every year.
The general reason that you shouldn't take a counter-offer is because a counter-offer can't address non-financial concerns and, depending on your manager, you might be viewed as a flight risk. In my case, my manager sincerely gave me his word that he'd never attempt to replace me -- he's been true to that. I don't know if you have that sort of rapport with the new CEO; I'd guess not since he's new. It also seems that your work is more project-based, meaning that once the project is finished, you're either re-assigned or cut loose. That would make your situation riskier than mine was.
Have you accepted another offer already? If so, this would be reneging on that offer. In my case, I hadn't signed anything yet.
Finally, just for what it's worth, although lay-offs are extremely hard to stomach, sometimes they are not a sign of incompetence. Just as a new CEO might come in and get everything wrong and fuck over the company, it's also possible a new CEO might come in and immediately see flaws and problems that the old CEO was blind to. If you stay, you should try to have a positive outlook.