Didn't you disable Connected Standby? If so your SP3 is coming back from hibernation not Connected Standby.
I turned it back on today, but it still doesn't look like it's doing the right thing. I think at some point I enabled hibernation through the local group policy. Does that have to be off for CS to work properly?
If you've disabled CS, you will be able to turn off the screen while keeping the device awake and run desktop programs. SP3 doesn't have a native function for this so you will have to download one of the various utilities out there that can do this. The one I've used in the past is Turn Off LCD, a very light-weight, no-installation-needed tool. You can click the utility or even pin it to start to instantly turn the monitor only off. (You might need to run it from an elevated prompt depending on your PC account's admin rights.)
I'll check that out after I confirm that CS is working properly on my SP3.