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:

elliptical orbit gif

I think the only way the moon could be getting further away is if the earth was losing mass or the moon was gaining energy somehow