The moon has just a little bit more speed than the Earth can hold onto

Unfortunately that’s not how orbital mechanics works :(

If the moon had instantaneously more speed than the earth can hold onto (e.g. more centrifugal ‘force’ than balanced by the gravitational force), the moon will accelerate up, until the forces become balanced. This makes a elliptical orbit, like this:

Apparently the reason the moon is getting further away is that it’s gaining energy from earth’s tides