Many people use following extension to backup chats (it only exports, but you can use the created .md files in new chat sessions or open in editor and copy parts you want.
Some users seem to have issues but they are often resolved easily. However you would need to be more detailed about the issue or what you tried or what errors you had incl. the chat ID which could help the cursor team debug the issue.
My experience with latest versions has become better and better.