Hello & Welcome to Just Answer,
Sorry I didn't see your question earlier and to hear of any issue you are having with your device.
I was looking for the procedure to replace the screen on the Oregon 600, and the "parts" to do the job, but unfortunately is was not able to find the replacement screen for this particular device.
Also, I found the instructions for replacing a screen on an Oregon 450, and I suspect the steps would be similar (Click here to see the Oregon 450 instructions). But without a replacement screen available, I would not recommend taking the 600 apart to do the job.
What I would recommend is that you check to see if repair is available via the Garmin "Out of warranty repair" page, which I have a link to below.
To check if repair is available - and the cost - use the pull-down selector on that page to select your device - to get a quote. If repair IS available, scroll down the page to see the steps needed to be able to have your device repaired.
I hope this helps. Please let me know what you find or if you have any more questions.