Level vs. Tier
Each DDAL module is designed for a particular Average Party Level (APL), and assumes a party of five characters. They also include instructions on rebalancing the module for a lower average level and a smaller or larger party.
However, it is Tier1 that determines which characters can play a DDAL module, not level. Tier 1 is L1-4, T2 is L5-10, T3 is L11-16, and T4 is L14-20. The D&D Adventurers League Player & DM Pack on the DM's Guild includes a Content Catalog showing which modules are appropriate for which tiers of play.
As for leveling cadence, as of Season 9 (the current season at the time of this writing), it's almost entirely within the hands of the players, not the DM. Characters advance at the end of each module (even 1 hour mini-modules) if the player wishes to. When running hardcovers, characters advance after a certain number of hours (8 hours, 4 in Tier 1), again if the player wishes to (though the DM can trigger advancement early in hardcovers).
If you're playing DDAL, you always use the current season's rules regardless of the age of the content, so you'll need to review the rules each season to see what has changed. When playing older content, you always disregard any leveling instructions in that content and follow the current rules.
Depending on the available content being run at your particular venue, players can simply decline to level up if doing so will take the character out of Tier for the next available module.
While trying to present a consistent storyline is a laudable goal, and each season of DDAL modules does have some consistency, that is not actually the design of an organized play campaign. It's meant for drop-in-drop out play. If you can have consistent players, great... but don't expect it.
I have played and run DDAL at a variety of different venues2. The best experiences I've had as a player are actually ones where a single person selects the content for the whole group. That one person decides what content all the DMs run on a given day. Usually it's two modules of different tiers, sometimes (like at the start of a season), all the DMs run the same thing. Most conventions with a strong DDAL presence run the same way - the content is decided at the event level, not the individual DM level.
1 There are a very small number of exceptions, most notably the introductory modules which consist of 1-hour mini modules, and are restricted by level.
2 I have five game stores within half an hour's drive, upwards of a dozen within an hour's drive, and currently run a private (but still DDAL) group online. I regularly attend conventions, as well.