I am a US Citizen that is employed and earns income in a foreign country. I am completing my taxes and think I screwed up as Turbo tax says I owe a large amount due to some triggering of the AMT.

I have a pretty simple life and limited investment income, so the question boils down to this:

Total of income earned is $300,000
Total US deductions (mortgage interest, etc) = $100,000
Total Foreign taxes paid = $77,000

The way I see it, if my maximum exclusion for foreign income is $80,000, that leaves me with a taxable income of (300-80-100) $120,000 - but I cannot claim the foreign taxes.

If I calculate without the exclusion, I have a taxable income of $200,000, but should be able to claim the $77,000 credit.

I have filled out so many worksheets and forms, I don't know which come into play and how the AMT has been triggered. Is there anything I can do to not have to cough up an additional $16,000.

You can claim foreign earned income exclusion of $87,600 and on the rest of the income you can claim foreign tax credit. That you can file both the Form 2555 and Form 1116.
The income you report on III6 is 200,000 minus 87,600 and foreign taxes paid is... use simple arithmetics.
Your U.S. Tax Return: U.S. Citizen or Resident with Foreign Income

Double check the AMT calculations. There is a provision that allows you to exempt the Foreign Tax Credit from AMT calculations.

You may want to consider getting professional tax help.

You should be OK... and although the calculations are tedious, I have seen exactly ONE brilliant paper explaining all the pros and cons of exclusion vs credit and working out the different scenarios: you must read it in its entirety: http://findarticles.com/p/articles/mi_6773/is_1_7/ai_n28522856/
It is called "Optimal decision between foreign tax credit and foreign earned income exclusion", International Journal of Business Research, Jan, 2007 by James G.S. Yang, Agatha E. Jeffers... complex, but completely blows away all the 'general' and 'casual' commentary on the subject.

The AMT can also be triggered by filing the F2555, the instructions require you to complete a worksheet within the instructions for the form 6251, to determine the impact.

Also, remember within the last two years the IRS can changed the way your remaining income after the exclusion of foreign income. This will cause your tax to go up. However, hopefully your credit can reduce it.