After I selected a different org in Org Picker, it keeps showing the previous selected org.
Hi @EstebanRomero84 - Thanks for reaching out. Can you share a bit more detail:
SFDX Authorize an OrgHi @smaddox-sf - Please find more details bellow:
Authenticate an org using SFDX Authorize an Org
Click the Org alias in the status bar to bring up the Org Picker
Click on another org that you previously authenticated
Thanks @EstebanRomero84 - On the VS Code version, can you share the version of the Salesforce extensions? You can see it from the marketplace tab - example:

Can you also copy & paste what you see in the 'Output' panel for the steps you listed above?
@smaddox-sf
Salesforce Extension Pack: 51.12.0
can we confirm that defaultusername field in .sfdx/sfdx-config.json file in your project folder is the username of the correct org (previous org you have authenticated in the step "Authenticate an org using SFDX Authorize an Org")?
defaultusername field in .sfdx/sfdx-config.json file is the username of the org I selected in step 3 (In my example it would be Cladd_Qa)
As I understand, the actual default org is set correctly, just that the org picker doesn't show the correct one. Is that right?
Yes, that is correct.
This issue has been linked to a new work item: W-9421700