Sharing this in case it helps anyone someday.
My vacuum pump was cycling after only being off one second, so I started testing the system to see where it was leaking. Eventually I determined it was the electrical solenoid. When it was at rest, the valve was not completely closing.
Since I didn't want to wait for a new part, I tried squirting a bit of grease-less lubricant into the two ports, and cycling the solenoid a few times by giving it 12v. After a few cycles it started holding vacuum. I reinstalled it, and now the vacuum pump will stay off for about 30 seconds.
Information from Dynomite's posts and Haibeck's web site were very useful for troubleshooting. Thanks, guys.
