You are right that it is very competitive these days. Our field - extremely competitive. What they were doing in Chicago over the last couple of years is actually cutting down the number of Ph.D. admits in Middle Eastern History to make our field 'healthier' (and this information actually came to me from a senior professor in UChicago).
Chicago is one of the very best places in the world to study the Ottomans. I mean ... it is Harvard and Chicago for the Ottomans - the big two. (Right? Everyone, right?) So if you accept that Masters degree, yes, it will be awesome and a great asset for applying to Ph.D.
Your professors are right about funding, of course - it is expensive. Fully-funded Masters in our field are very rare though. If this is your only offer AND you are totally committed to this ambition of becoming a historian - then, perhaps, you should go for it.
But, perhaps, I am talking rubbish - you need many different opinions on this. This is a big decision you are making.