A career change to professional flying at your age can be achieved. But those who succeed are few and far between, and tend to be somewhat celebrated on these pages! Indeed, our own Capt Pprune was a late arrival to this field at 37.
However, I would echo the cautious note of the Capt Pit Bull and Maximum. The job market at the moment, and for the forseeable future, just does not favour a late entrant with little experience. That will change, in time, but it's difficult to predict when, and how long the next period of 'good times' will last.
I wouldn't burn any bridges just now. Maybe take the severance, but get yourself set up in a new IT position while you obtain the ATPL qualifications in your spare time. Look at the aviation market when you are fully qualified and only then make the decision whether to leap or not.
And, as I've often said here before, make sure your family fully understand the risks of your chosen course, and are prepared to back you for the long haul ahead - even through unemployment and potential homelessness. Believe me, the risks are huge!