Note that in godot versioning scheme major releases are practically guaranteed to include compatibility breaking changes and while rare, even minor releases can occasionally do so. Latter tends to do so when fixing a bug or unintended behaviour. And breaking changes are usually documented in release notes.
While it's nice that the core developers try to include some resource conversion capability so users can more easily port their projects between the major releases if you are considering porting your project you should always be prepared for having to get your hands dirty in fixing or even outright rebuilding some aspects of your project and/or related tools.
That is to say the decision to port your project from one major version to the other shouldn't be made lightly. I would also add that the best practice would always be to make a back up/copy of your project first.
With all of that out of the way... I'm not sure if the resource conversion is intended to be capable of converting AnimatedSprite resources, if so then perhaps something is just broken/bugged and you might just have to wait for a fix. Or if you can't afford to wait for a fix that might never come then it's probably up to manually rebuilding those entities/scenes that use AnimatedSprites.
