I amended my taxes and now owe money, but my original tax refund is still on the way. Do I cash the check and have them debit my account? Or ignore the check?

Yes, go ahead and cash the check and then let them debit your account for the balance due on the amendment. 

When you amend your return, the original and amended returns are processed separately. The IRS doesn't net your results until well after tax season is over. 

Hopefully you entered the amount tax refund you will be receiving on the amended return so that you get the correct amount of taxes owed on the return.

You should really wait for the tax refund before sending the amended return since the IRS may have made changes to your tax return which would affect the amount of tax refund you will receive.

You can send the amended return now with your payment.  Once the tax refund arrives you can cash the check.

You should not have amended until you got the first refund.  You will still get the first original full refund (and keep it) or need to pay the first tax due.

The amended return will only be the difference you get back or need to pay. You can confirm this by printing your amendment and looking at the 1040X. Line 18 should be your original refund amount and line 21 should be your additional refund. If you paid on your original return it will be on line 16 (but don't include any penalty on 1040 line 38). If there is an amount owed with the amendment, it will be on line 20.