My question is will I run into any problems in the airport by booking this way?
The only problem is that if you book separate tickets the airline is not liable if the first flight is canceled or late causing you to miss the 2nd flight on a separate ticket New York to Paris. This means you would have to buy new roundtrip tickets New York to Paris.
If you gamble buying separate flights to save $1300, give yourself at least 6 hours to make the New York to Paris part or fly in the day before and stay a night in NYC if it does not cut into the saving to much
Or if you are trying to find similar savings you might look at booking the transatlantic part separate from the inter-European flight part; for example, Houston direct to Frankfurt or London or Amsterdam.
Once you get over there, there are a lot more options to get to Paris by flight or rail or bus