Is the new food low calorie, for weight reduction perhaps?

When they vomit foam / saliva, it is because they are over hungry and the stomach acid has built up. The cat is going too long without food. When they eat and throw it all up right away, it is that they are over hungry and just wolfed the food down without proper chewing. If you suspect the cat is overhungry, it is best to give a small amount of food, wait 10 minutes, then give the rest of the meal. So the stomach has time to realize it now has food. And the cat can realize he is not "starving to death" so he can eat the rest of the meal without wolfing it.

Don't know if this helps?

kb2jyx mentioned eating more, urinating more, and not gaining weight - could be diabetes.

A slightly more common ailment in older cats is overactive thyroid. The metabolism is running really high. So the food gets burned off faster and meal times need to be more frequent. When it gets to the point the cat is losing weight, that is usually when people see the vet. A blood test diagnoses the over active thyroid, and a simple pill a day solves the problem.

Has Barkeley recently had any teeth removed (also common in older cats)? That could explain why he needs to eat more often too, if he has some sore gums just now.

Hope something here gives you some ideas!!
Sandra