Hello, and welcome to the Apple (user-to-user) Community!
Re: "Cannot sync the airpod pros with the different case."
We do not know which AirPods Pro model you have in order to help:
If needed: See how to: Identify your AirPods - Apple Support
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Below: Check for the correct AirPods Pro model in order to follow the step-by-step guidance in the article below
See Apple Support's online article: How to: Set up your replacement AirPods or charging case - Apple Support
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Re: "Case has a blinking orange light when the case opens".
Generally means "error" : AirPods would usually need to be reset and paired ...
(yet your case being a different one. may be that error )
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If needed: Apple support's video will show you how to re-set:
https://www.youtube.com/watch?v=40qlsGWEadw
All the best :-)