For goodness sakes don’t swallow exceptions! It’ll kill you (not really)!!!
Don’t try to hobble along. If an exception is thrown, log it, present the user with a “Sorry, but we had a problem…” message and assure them you are on top of the issue and graceful close the application or restart it.
In business programming there is no reason to try to “Save” the context. It’s a disaster waiting to happen. By swallowing an exception the application is put into an unknown state. Badness can happen when your application is an unexpected state. Anything from data corruption to your application being compromised. One of the strategies of hackers is to test edge cases, looking for an in or an error to gain more information on compromising the application.
When your application throws an exception, logged it and close the application. The context may have been lost, but the application lives on.