Is there any way to fully clear a stormcloak/imperial camp?

The Unofficial Skyrim Patch (original / Legendary Edition - PC) / (Special Edition - PC) / (PS4) / (Xbox One) - should fix that issue (it's actually part of a quest that was not configured properly):

• The quest built to deal with the essential civil war camp soldiers was not configured properly. The script intended to attach to aliases for Tullius and Ulfric was never attached because the aliases had never been created in the proper place. This should now be corrected and this quest will be started again if it's not already running.

• Imperial Legates and Stormcloak Commanders in camps throughout Skyrim will no longer remain unnecessarily essential after the Civil War questline is completed

USKP full changelog

You're correct in assuming that the immortal NPCs are quest givers for the opposing faction. Unfortunately, even though you've already finished the Civil War quest line, those opposing NPCs are still immortal.

If you're playing on the PC, you can use the console to remove their invulnerability: Press ` to open the console, click on the NPC, and type SetEssential 0.

So, galacticninja's answer is correct that the USKP fixes the problem where the commander's of the opposing faction's camps do not have their essential flag removed when you've completed the civil war quest line, but, honestly, having played with that, I still felt it was, at best, underwhelming. It still didn't really feel right, as the soldiers just randomly respawned and there was no way to truly deal with those camps once and for all.

Well, the modder Korodic addresses that with the mod Civil War Aftermath. A great overview of that mod is provided by the YouTube personality TheEpicNate315 in this video at 5 min 36 sec. A short description is that the mod allows you to kill all the people in the camp AND burn the camp to the ground.

This mod is available on Nexus for Skyrim SSE and Oldrim and on Bethesda Net for XBox One.