All DVR recorded shows start 1 minute early and ends 1 minute early

Done.  I just emailed info and link.  Note same problem has been reported in the past, but I didn't see what the resolution was if any.

I'm not sure if this is the total answer but here is what I have found out with help from Cox  (thank you).

One of the two DVRs in our home had "essential DVR" option which allows only 2 simultaneous recordings.  The other has "preferred DVR" option which allows 6 simultaneous recordings.  I think the problem existed when for example you already have 2 programs recording, but then a higher priority series recording starts a 1 minute earlier, then cutting off the lower priority program prematurely.  That's my theory anyway.

You can turn off the "auto pad recording" option this way (thanks Cox for this info):

Press the Contour button on your DVR's remote control and use the arrow button to scroll to the Settings menu (gear icon). Press the OK button on the remote to select Settings. Within Settings, scroll down to Preferences and press OK. Then press the OK button to select the General menu. Scroll down to the bottom of the General menu to "Auto Pad Recordings." Is this option current On or Off? If this option is enabled, press the OK button to disable/turn off this feature. You can then exit the Settings menu.

I have upgraded the essential DVR 2 option to the preferred  DVR 6 option.  Now monitoring to see if that works as intended now.

After various strange problems, the DVR wouldn't boot and Cox just replaced it with new one.  At least everything is working again.

A possible reason your cable box won't reboot is because it's defective.  If that's the case, your only option to fix it would be to replace it.  However, it's more likely caused by a signal issue.  A cable box won't reboot without a good signal.  The lag with remote response is another indication you have a weak signal.  When you enter a command with the remote, the command is sent to a Cox server and your cable box waits for a reply.  A weak signal will cause delays.

You should schedule a tech visit to check your signal strength.  Be sure to have it checked at the tap to determine if it could be an issue with the main line.  If that's the cause of your issues, you won't be charged for the tech visit.

I spoke too soon.  Cox replaced the DVR that wouldn't boot, replaced an amplifier & filter in the main line connecting cable service into our garage.  New DVR booted fine, but a day later I noticed again that all scheduled DVR programs start a minute early recording.