The 7.0 version of the AL Player's Guide has a slightly different wording:

As D&D Adventurers League play focuses on creating a welcoming, fun environment, neutral evil and chaotic evil characters aren’t allowed.

As an Adventurer's League player, I'm hugely in favor of this. I'm giving up my Saturday or weekday evening to join together and go on an adventure.

The nature of the structured, drop-in format means that these sessions are generally "go through and solve a pre-written module" rather than "it's the DM's world — or the table's world — and any module content is simply a behind-the-screen aid to reduce prep time". This means there's less latitude to explore alternate paths and still, you know, get through to a satisfying end. Adventurer's League modules aren't written with evil characters in mind, and by going against that, you're wasting everyone else's time.

Additionally, unlike a home (or school or workplace or whatever) gaming group, you're not necessarily my friend. In a gaming group, it's important for everyone to be on the same page, but since in public play there's no preexisting social dynamics to work that out, cooperation with the alignment rule is much more important. In Adventurer's League, it's expected that as a team, we're the good guys.

So, if you show up and act out of coordination with that, you're hurting the fun for other people at the table. And since they don't necessarily know you well enough to feel comfortable objecting, they may just have their fun ruined without any real recourse.

In a home game, you might accept the alignment change and then ask for some way to find atonement. In AL, ain't nobody else got time to make this thing all about you.

So, yeah, the appropriate thing to do in example 1 is retire that character from play. Make a new one, and make her or him a hero like Adventurer's League asks for. (Or, as the exception lists, be lawful evil but with a bond which ties you to getting along nicely with others and doing the adventure.)

In example 2 — lawful evil — you could have the added option of switching factions. I don't see any reason why the DM wouldn't allow this. You loose all existing renown and benefits from your current faction, and starting with no reputation or rank it's probably fair for you to not get any in-game faction benefits this session.

The section of the guide on Factions has this further guidance, which underscores my basic point and should definitely be kept in mind if you decide that a faction switch is the way to go:

No Undermining Other Characters. Adventurers are brought together by common cause, and they’re expected to work together to overcome challenges. Though certain factions might find others distasteful, the characters must put that aside and work as a team.

The only exception to retiring or changing the character would be if you feel the alignment change is unfair — you weren't acting out of an evil impulse, but an accident happened, or something. In that case, talk to the DM and ask to a) not have an alignment change after all or b) accept it for the session, but promise to actually behave and cooperate, and figure out what to do next when the session is done. If you can't work something out with the DM, talk to the organizer.

The second option, with some caveats.

While Dale M's answer is technically correct, we should also consider sentient weapons' traits mentioned in DMG (p. 216):

A sentient item has a will of its own, shaped by its personality and alignment. If its wielder acts in a manner opposed to the item's alignment or purpose, conflict can arise. When such a conflict occurs, the item makes a Charisma check contested by the wielder's Charisma check. If the item wins the contest, it make one or more of the following demands:

[...]

If its wielder refuses to comply with the item's wishes, the item can do any or all of the following:

• Make it impossible for its wielder to attune to it.
• Suppress one or more of its activated properties.
• Attempt to take control of its wielder.

So, while a character of any alignment can try to attune to the sword, a sentient sword might try to actively resist it, if the new wielder is not up to the weapon's taste.