Level vs level
Each DDAL module is designed for a particular Medium Group Level (APL) and assumes a group of five characters. They also include instructions on rebalancing the module for a lower average level and a smaller or larger part.
However, it is level1 which determines which characters can play a DDAL module, no level. Level 1 is L1-4, T2 is L5-10, T3 is L11-16 and T4 is L14-20. D & D Adventurers League Player and DM Pack DM's Guild includes a Content Catalog showing which modules are appropriate for which levels of play.
As for the rate of upgrade, from season 9 (the current season at the time of writing this article), cin the hands of the players, not the DM. The characters advance at the end of each module (even mini-modules of one hour) if the player wishes. When performing hardcovers, the characters advance after a certain number of hours, again if the player wishes (although the DM may trigger the advance in hardcovers earlier).
If you play DDAL, you always use the rules of the current season whatever the age of the contentyou will need to review the rules each season to see what has changed. When playing older content, you are still unaware of the upgrade instructions in this content and follow the rules in effect.
Depending on the content available on your site, players may simply decline to upgrade if it allows them to exit the character level for the next available module.
Although trying to present a coherent scenario is a laudable goal, and each season of DDAL modules has some consistency, it is not really the design of an organized game campaign. It is designed for the game drop-in-drop out. If you can have consistent players, all the better … but do not expect it.
I have played and directed DDAL in various places2. The best experiences I've had as a player are actually those where one person selects the content for the whole group. This a nobody decides what content all the DMs work on a given day. These are usually two modules of different levels, sometimes (as at the beginning of a season), all DM work the same way. Most conventions with a strong DDAL presence work the same way: content is decided at the event level, not at the individual DM level.
1There are a very small number of exceptions, including introductory modules that consist of mini modules of one hour, and are restricted by level.
2 I have five gaming stores within a half-hour drive, over a dozen to an hour drive, and I currently manage a private group (but still DDAL) online. I also regularly attend conventions.