[big gear head]

A 3/8-16 helicoil will have 16 threads per inch on the outside, not 14.

The reason you will not find an adapter like you want is that the minor diameter of the 7/16-14 thread is .360 and the major diameter of the 3/8-16 thread is .375. If you try to cut 3/8-16 threads inside a 7/16-14 bolt the 3/8-16 threads will actually cut through the root of the 7/16-14 threads. I just tried this to see if it would work and it didn't. I took a picture, but couldn't upload it where I am right now.