If you create a new profile and start a new game, do you have any problem with these delays in combat or dialogue? What about reloading an older save that was working before (or if you load a save that had minor problems, is it still minor or comparable to your recent saves)?

If a new game, or older saves, display the same issues as your recent saves, the problem may be with the game itself, or be system related.

Are you shutting down all non-essential programs (especially anti-virus) before starting the game? Firewalls have caused conflicts (generally on loading screens, though) and overlays from graphics tweaking/monitoring programs or chat programs have also caused issues.

Try verifying local files: in the Steam library, right click on Divinity Original Sin Enhanced Edition and select Properties, switch to the Local Files tab and then click on the 'Verify Integrity of Game Cache...' button.
With the GOG version, in the (optional) Galaxy client, select Divinity: Original Sin Enhanced Edition in the left column, then click on the More button, and in the Manage menu select 'Verify / Repair'.

If applicable, disable Steam cloud support either globally (in the client click on the Steam menu and select Settings, and then Cloud) or just for this game (in the library right click Divinity: Original Sin Enhanced Edition and select Properties, then switch to the Updates tab and check the Steam Cloud section).

Next, try browsing to the '..\Documents\Larian Studios' folder and rename the 'Divinity Original Sin Enhanced Edition' subfolder.
This folder contains the saved games, configuration files and a level cache folder. Deleting or renaming it will get the game to recreate it on startup; playing the game from a different Windows user account would effectively do the same thing. With Steam cloud support enabled, Steam would just download the cloud copy of your existing profile.
After that, start D:OS, create a new profile and maybe check a new game. Exit and copy a couple saves from the renamed folder into the newly created profile's savegames_patch folder. If that lets you load the saves and play without issues, move the rest of the saves over.
If that doesn't help, delete the new My Documents D:OS EE folder and rename the original back again.

If a new game is fine and the issues are consistent in your existing saves, the problem is more likely something corrupt in the save itself.

Please email supportdos@larian.com, with a description of the problem (or link to your post) and the report.zip file (or a Dropbox or Google drive link, etc) generated by the D:OS EE Analysis Tool (for Windows).
Extract the zip file somewhere and run AnalysisTool.exe. When that loads, browse to the install folder (if it is not automatically detected), optionally run the tests first, and then generate a report (it will be saved in the same location as the Analysis Tool).

The report will contain system and game information, as well as your saved games. To reduce the file size, you can delete most of the saves from the zip (double click the zip file to open it in Windows Explorer, then browse down to the savegames and savegames_patch folders); keep your latest save, and maybe the most recent working save and oldest that started having issues.

For OSX, saves can be zipped by right clicking the folder(s) and selecting Compress, in the folder:
~/Documents/Larian Studios/Divinity Original Sin Enhanced Edition/PlayerProfiles/<ProfileName>/Savegames_patch

Linux is similar, with the saves in the home directory.