Congrats on finding some morels... sounds like you may have several there in your area.
Here at my place, they are thin, and short lived... here it sort of goes from winter to summer... often and we don't have a long drawn out spring.
I think that makes our morel season really short... usually that first week in April (here when dogwood blooms first start opening) is about the only time I can find them. I have gone back and looked again and again several times, to find NO more.
I think they will last in the woods a few days, perhaps even a week, if the conditions are right, cloudy, moist..
But if it turns hot and sunny... from my experience they start drying out pretty quick. Just a couple days and they will be too dry.
Good Luck - hope you find a pile.
On cooking them... if you are ok with eating flour...
Put a 2-3 table spoons of olive oil and butter in your skillet and get it frying hot (med high)...
Prep your morels... I usually just slice them in half from top to bottom, and wash good, and dry them off some with a paper towell.. but leave them a bit moist.. then in a bag with some flour and shake them up good to coat with flour well.
Then fry them up in that butter/Olive oil mix... and salt/pepper to taste.
Good Stuff.
TNHunter