Go to job interviews to change jobs, not to match what they offer.
Generally speaking it is a bad idea to go to job interviews then come back with a letter saying they're going to pay me X dollars, and I want that amount from you. They'll likely not agree with that approach and if they do, it usually means they're caught off guard and will likely dump you later on when they can safely hire someone and swap you out.
Best idea is to go to job interviews, find a position, turn in your 2 weeks, perhaps hear what they say, but I would lean towards going to the new job. After all you already made your demands, so would you trust an employer who suddenly able to meet your demands after saying no when you're leaving? Why couldn't they simply give you what you ask when you asked for it? You got to think of it that way.
The only time I would sort of agree with hearing their demands is if you never asked them and they make some offer that seems good. For example, you turn in your 2 weeks, they say hold on will giving you a X% raise be helpful? However, if you made your demands, and they said no, or gave you a generic excuse, I would pass on the offer and continue with the notice period. For example, you say, "I want a 15% raise." They say, "Sorry, we just don't have the money right now." Then you say, "Okay, I found a new job, here's my 2 weeks notice." Then they say, "Hold on, would the 15% raise make you stay?" Then at that time, I would say no thanks and continue with my notice.