Can I go to the ER for a potentially cancerous mole?

Dermatologist will exam it, take biopsy. If it's deemed none cancerous, the doctor will probably remove using freeze technique.

The ER will tell you to see a dermatologist, but it may take 20 hours for them to get around to doing it, 

People working in the ER are far too busy with people who are really sick to stop and tell you that the ER doesn't do that. After ten years another couple of weeks is unlikely to make much difference.

Your concern is pretty common. A new product on the market is easy, more accurate than a biopsy. A patch is put over the lesion, left in place for several days, then removed and sent in. Much easier and skips the dermotologist, unless it is positive.

Made by a company called Dermtech.  Perhaps you can make your provider aware of this option and they can order it for you!

https://dermtech.com/

You can, but they will tell you to see a doctor or dermatologist.

Cancer is not diagnosed at the ER.

You will just be wasting everyone's time, yours included.

No they wouldn’t. There are no derms that would come to theER for that, and an ER doc wouldn’t biopsy that either. And you would be billed a lot to be told that, a week wait won’t make any difference in outcome or treatment. Be patient.

they'll give you the same referral your doctor will,the only difference is that they will charge you $10,000 versus a couple hundred from your doctor