keeping up with organic nature of my campaign party, doling out XP while
not difficult mskes advancement feel too slow. Does anyone see a problem with simply advancing the party the first few levels with each with the conclusion of each "chapter" of the adventure module. Or base the split of XP on number if sessions attended.
are there other XP AWARD methods have worked as well?
I also don't see any problem with your idea (which I'd call "story-driven XP"), just go for it!
There are countless of methods and variations thereof for awarding XP in a way that's different from combat-based or treasure-based.
For instance one totally "meta" method, is to grant XP based on when the players need, rather than the characters. With this method, the DM agrees with the players about how long they should stay at current level to get the most "fun" out of it (of course some players have most fun in levelling up...) i.e. stay at current level until they had enough time to use all the new stuff they gained, got enough chance for some tactical understanding and combo experimentation, and haven't become bored yet. Then, agree together it's time to level up to get something new to do.
Similar to story-driven XP but more complex is when instead of advancing automatically by the chapters, you bound the XP to actually solving pieces of the plot, discovering mysteries, win allies etc. This normally also includes defeating enemies of course, whether by combat or some other way.
BTW you don't have to completely phase out combat-based XP, but you can easily choose how much each source of XP should matter. For example, I player for a few years in a campaign where the DM was giving XP both for straight combat and for quests, but random encounters were worth half XP compared to combat that was related to an active quest. This house rule was meant to discourage "XP harvesting" and instead encourage focusing on your quests.