Welcome to FT and congrats on your first post.

AA opens awards seats on just about every flight as long as there are seats. The caveat is that they are EXTREMELY high redemption values. To get the lower point-value awards, you need to do some work...

Don't pay anyone a penny. You can do it for $9.99/month. The old adage that seats open up at T-331 days isn't fact any more. Airlines hold back seats for sale, and often never open any premium seats for travel if it's a historically high demand date. In your case, going to Europe in Sept/Oct is much better than trying for something in July. Calling AA will also not get you a seat any sooner.

For $9.99, you can get an ExpertFlyer subscription (web site). This will let you search the "Awards and Upgrades" inventory on AA and other airlines. you can select "U" inventory if you are looking for business class. you can search multiple dates and, if you see U0, then you can set alerts to let you know if any open up. You can also search "C" inventory if you want to pay coach and upgrade with 25,000 miles + $350 each way, per person. When you search, it's recommended to look at the long haul segments. Don't search something like DFW-FCO that may result in DFW-PHL-FCO, because the inventory result may not be accurate. Be sure to search with the 'no codeshare' and 'nonstop' options to show you what's most relevant. Be flexible and search other gateways like MIA, PHL, JFK, ORD. If you find inventory on those routes, then consider flying the intra-US leg to get to the long haul. Lots of threads and advice in this forum. Good luck.