Do I have to claimed Home Energy Credits from previous years if the credit was applied to a different home that I don't own anymore?

The credit is a lifetime credit not one applied to a specific home. 

Yes.  It is a lifetime limit per person, not per home up through 2022.  In 2023, the lifetime limit goes away. 

For the section 25C credits, there is a $500 lifetime limitation (lifetime of the taxpayer, not the home). 

Section 25C includes things like doors, windows, insulation and efficient heat pumps, furnaces, A/C and hot water heaters.

Starting in 2023, the lifetime cap is removed and replaced with a $1200 annual cap.  Unfortunately, improvements that were made in 2022 must be reported on your 2022 under the old law, you can't defer it to 2023.