Hiya!
My idea would be to add-in some sort of "level rating" for being a member of "politics, society, religion, family, and all that stuff". I know there are the 'rules' for Factions, but I'm thinking something a little more in-depth. Something that gives the Players a means of attaining "points" for leveling up, and upon gaining a new 'level' in something (church hierarchy, clan leadership, etc), they get some kind of bonus/perk based on their Class.
IMNSHO, you can't "just remove dungeons" from D&D and expect the game to run as expected. The majority of the drive and XP system is based around "adventuring"....which basically means killing monsters and taking their stuff, usually in a dungeon. So you would NEED to replace that game-mechanic reward system with something else.
Gamma World 3rd (and 2nd edition, the only two I'm really familiar with) had your regular "Rank" (PC level) but it also had a "Rank" for your characters Community. You gained "Status Points" that indicated how favoured, trusted, and important you were to your home community, or any community (you had a separate 'status rank' for different communities). The advantages of Status in a community was that you could get help, borrow rare equipment, get free supplies/food/water, and generally be more influential in the community. It really did add a rather important part to a post-apocalyptic setting where even GRASS wants to teleport into your stomach to kill and eat you! It made Players actually develop feelings for a village, town or even outpost. The NPC's in them became important to the Players and their PC's.
Something like that would need to be done to replace the "DING! That last ogre put me over! I'm now 5th level...sweet! New spell coming!". In stead of "personal power gain", it would need to be "outside influence gain", so that the Players and their PC's become invested in WANTING to help or oppose various 'groups/communities' in the game world. Otherwise, they would feel like they don't matter and are just, well, murder hobo's. Litterally.
^_^
Paul L. Ming