How can I restore an Application state from a cloned system?

I successfully restored my iMac after a hard drive failure, but the Time Machine backup was a bit old.

I can access the "failed" hard drive as read only, which contains my most current files.

I would like to restore some applications to the state they were in when the hard drive crashed (Safari/Firefox/Chrome tabs, Stickies...).

In this answer, the answer suggests replacing the whole Library folder, but it sounds risky or long for just a couple of apps to be restored.

Where would I find these application states?

It depends on the app:

Non-Apple applications states or data can be found in ~/Library/Application Support/ usually.

Copying the folder named as the app to the respective path in your new system should restore most things.

This works for Firefox and Chrome.

It's also adviseable to copy the content of ~/Library/Saved Application State/.

Some Apple apps have their own folder in ~/Library/ like Safari and Mail. To get the Safari data and especially all your mails (in Mail), copy those folders. Others have their states/data in ~/Library/Group Containers/ or ~/Library/Containers/.

Very important:

Copy the folder ~/Library/Keychains to your (new) desktop and import your passwords. If you are also using a password app like 1Password, copy the respective folder in ~/Library/Application Support/.

Copy all preferences files ~/Library/Preferences/. Also check for invisible files and folders in ~/, ~/Preferences, ~/Library, /Library or /. They often contain either config files (e.g. .ssh) or even auth files or serial numbers for apps you purchased in the past.

You neither specified a system version nor an app version. Depending on your system or app version and purchase type (App Store or conventional) paths may vary.