The adventuring party with a consistent membership and unified goals is the most pernicious construct to good gaming known. Get rid of them. As a baby step, let's start with consistent membership.
The best gaming group I've ever known was the one with the most character versus character conflict. This CvC was driven by the existence of a large roster of PCs played by the same 3-4 players. The character roster started off small, and then gradually grew as people wanted to play new characters, as characters departed temporarily to pursue their own and then circled back when they needed the other PCs as allies. This created a dynamic, changing cast with no need to rely on player sentiment and niceness to temper dramatic conflict. It also allowed us to do things like kill jerk characters without the player running them feeling like we were "kicking them out of the group" or something, since they would have two or three other PCs hanging around who were woven into the story. Or characters who were or became incompatible would split off to pursue their own goals and accumulate a group of allies more amenable to their tastes. Every so often, something would change, or the different groups would come into contact and they might trade members, as priorities shifted or temporary goals caused different bands to ally.
I cannot convey in mere words how captivating this was. It allowed campaigns to feel truly epic, allowed time to be spent exploring the subtle mysteries of the world or facets that otherwise would have to be left by the wayside due to more pressing concerns, and allowed characters to act on their passions and beliefs with a freedom that an artificial commitment to avoiding trouble would constrain and dampen.
For example, I played in an Iron Heroes game in 2007-2008 that was the single greatest campaign I have ever been part of, at least partly due to the dramatic freedom this kind of play allowed. Our first set of PCs were the three scions of a powerful noble house. I played the youngest of the three brothers.
Each one of us had certain tendencies and inclinations, often conflicting, but we were brought together despite that due to fraternal obligation and fondness. However, our goals would frequently separate us across the kingdom, or one of us would be incapacitated for some reason while the other two were able, etc.
At first, we began by fleshing out and playing what would otherwise be NPCs. For example, the oldest brother once took some of his guards into the slums of the capital where he was ambushed. The other player and I, rather than sit this out, took over, named, and fleshed out two of the more important guards accompanying the oldest brother- Doc, a respected veteran, and Sgt. Fusker, the head of the detachment. These characters remained even after this incident (having acquitted themselves bravely), and whenever one of the brothers wasn't around, we would play them instead.
Similarly, the middle brother at one point ran off from his responsibilities to become a mercenary adventurer. He met up with one of our shifty uncles, and a mercenary that uncle had hired, once again played by the other two PCs. Even after the middle brother departed, the uncle and the mercenary (now his trusted bodyguard and assassin) remained and found a third person, a young impressionable nobleman, to accompany them on their own adventures. As people died, others stepped in to take their places, often transferred over from less active clusters who the currently active group encountered. For example, when my noble brother died, the uncle's bodyguard was sent to assist his nephews and keep the remaining two safe. At other times, there would be conflict, as when one group of PCs, a group of revolutionaries, captured said noble brothers and mercenary and were debating whether to execute them (in essence, we held an IC debate about whether we were going to kill our own characters, albeit ones we weren't playing at that moment).
This kind of natural proliferation kept interest high, as you never had time to get bored of a character before you were creating and playing a new one, and you started to long to find out what characters you hadn't seen in a while were up to. This proliferation culminated when the noble brothers provoked a civil war and then left the continent. We played out the civil war through three different parties in three different locations, switching between them each session and seeing the war through its various facets.
I give full credit to the referee for knitting all of this together and making it coherent. It was one of his best realised settings, and the consistent vision of its development and change helped keep everything synchronised. It helped that only a few points was there any sort of direct "quest" which would save the world. For the most part, it was a power struggle driven by personalities for control of the kingdom, something important but not so important that nothing else mattered.
Multiple PCs keeps people from over-investing in a single character, which prevents them from feeling like their goal in playing the game is to have that one character succeed at everything and be the most important person in the world. Preventing over-investment is critical to pulling off dramatic conflict and by extension, character versus character conflict, as it prevents people from getting into mindsets where they confuse in-character disagreement with out-of-character disagreements and feel that someone is picking on them or bullying them or actively conspiring to keep them from being cool or whatever.
Give this a try if you haven't already. It's actually even easier to pull off in settings where the PCs are disconnected murderhobos, as such groups IRL (gangs, essentially) had and have an ever-changing membership motivated by mercenary concerns and prone to violent resolutions of even petty disputes.