Even if you complete the quests the markers stay on the map, is this supposed to happen?

Quests completed before installing Questie also are stuck on the map.

If you're upgrading to the new version you need to perform /questie clearconfig

This will create the new keys in your config that the new quest cache system uses.

Please test again with the new 2.67 release. There was some zone map data fixes and a fix for EQL3 that seems to be causing some headaches.