When filing my 1040, I inadvertently put down an estimated tax payment in error. Should I wait to see if the IRS will adjust and me a reduced return or file a 1040X?

If you know about the error, I would file an amended 1040x and see if that results in taxes that need to be paid. That way if taxes are owed you are able to pay that before the tax deadline and not accrue penalties or interest on that. 

You should not try to amend your return until the original return it has been fully processed and you have received your refund or your payment has cleared.
Amended returns do not “catch up” to your original return and replace them; they are processed as two separate returns.  
Amended returns have to be printed and filed by mail.  It can take the IRS up to 12 weeks or longer to process them.   If you are due a larger refund than on your original return, your amended return should only show the difference and you will receive a separate check for it.  If you owe money on your amended return, it will show only the new amount owed, you will have to mail a check with the return. You can check the status of your amended return by clicking here: https://www.irs.gov/filing/individuals/amended-returns-form-1040-x-/wheres-my-amended-return-1

 Click here for amending instructions: https://ttlc.intuit.com/replies/3288565