Well one thing to try that won't cost a whole lot is to change the oil filter. Also if you go to John Deere - Parts Catalog . There is some kind of a valve in behind the filter.
First time I ever noticed it, someone said to replace the filter. I too thought it was due to the cold oil, but I forgot to check it after warming it up. I sure hope its not the oil pump going bad.
So let me get this right, we have not tried a new $4 filter yet? I would say like DP check and verify the oil pressure. Oil pumps seldom go out and when they do its on high hour machines and ones that are severly neglected where the filter has been bypassing from being plugged up for long periods.