Hi JA-E,
welcome to the Apple (user=to-user) Community!
Re: Pairing airpods with new charger
If you haven't seen it before, Apple support has an article onine (link below) that may help:
See how to: Set up your replacement AirPods or charging case - Apple Support
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Re: The light on the charging case never turns from clinking amber to blinking white
and my iPhone never "finds" it as a device in bluetooth
Issue may be resolved by setting up the charging case following the steps in the article above.:
Else if more help is needed:
blinking amber shows there is an error: may mean AirPods need to be reset and paired.
How to reset your AirPods and AirPods Pro - Apple Support
IPhone needs the AirPods to be in discovery mode (blinking white) in order to find them,
if you still have an issue after setting up the new case, resetting (as above) will likely help,
or
You might want to try wearing your AirPods, and on iPhone:
Try Settings > Bluetooth > tap "Not Connected"
All the best :-)