I've done this Sir. Going from Standard to Legendary Edition to save money on DLCs. With a Level 35 character.
I would:
- Back up your save files and put them somewhere safe.
- Ensure that when the 'overstall' takes place that Steam is reading from the disk and not trying to download the whole blinking game again.
Your DLC's will chain install automatically.
There are aspects of Steam that aren't particularly intuitive. e.g. What DLC do I have installed? and Where do I enter my redeem code? I had to Google that.
If you run the default Skyrim Desktop shortcut following 'overstall' and look under 'Settings' (check this reference, I don't have it open in front of me and use SKSE at game launch) then your DLC will be a series of selectable checkboxes under the main Skyrim.esm file.
Check the DLC you want to run at any time, as they are not main quest dependent, and you're good to go.