You don't need to move any files or anything.

After submitting a track to the workshop, you (and whoever else) need to update their subscriptions (typically just hit the workshop button from the main menu, close the steam overlay, and accept the dialog to allow it to refresh; alternatively close/reopen the game). After the subscription is updated, your (the track creator's) level select will display the newly downloaded tracks at the very bottom typically. Between the official tracks and the community tracks are the tracks that are in your (track creator's) "My Levels" folder. If you're playing online, don't select those; rather select the one downloaded from the workshop at the very bottom of the list.

The reason your level editor tracks appear in the actual gamemode levelselect is so you can properly time moving/rotating objects when initially playtesting your track. The timing is off when you use the in-editor "test" (~ tilde) mode because it allows you to start moving during the "3, 2, 1" countdown.

Just make sure you subscribe to your own content, update your subscription, and select the "downloaded from workshop" version of your track rather than the "in my editor folder" version.

Originally posted by ElectroMite:

After creating a map, I uploaded it to the workshop.

I've done this a few times and in order for me to be able to play this online with other people is to have my local map files removed from it's original folder so the client can download from the workshop file instead of reading it from my local file. Is there a way to do this without moving or deleting my local files.

If I delete my local files, this means I won't be able to update my current map and I have to republish the map.