I can see one flight from ORD to DXB with RJ on 12/22 but nothing to AUH on either date. For the return, there are six seats available on BA from AUH to ORD (via LHR of course) on 1st January, and the same from DXB. Nothing on the 3rd January it seems.

It can be challenging finding long-haul redemption flights. It can help if you're flexible with your dates. As you point out, the additional carrier charges (especially from the USA) when redeeming Avios on BA for transatlantic flights can be a real disincentive. Equally, the recent significant increase in the number of Avios required for domestic redemptions on AA and AS has resulted in a devaluation of the value of Avios. However, they can still be good value especially when cash fares are high and you also have the ability to cancel for a full refund of the Avios up to 24 hours prior to travel.

For partner redemptions you are totally at the mercy of the partner carrier to determine what, if any, seats are made available to those booking with BA. It does seems to be feast or famine, but availability does vary and a flight with no Avios availability may have some a week later - and the opposite is just as true. For domestic US redemptions where the cash component is very low I would strongly recommend booking flights if you see there's availability on the route or date you are interested in. If you need to change, you lose the cash component but given that it's usually $5.60 for a economy redemption on AA that's basically peanuts.

For both my recent EU vacation (or holiday, as I prefer to call it) plus my upcoming Christmas trip to Chicago with AA, Amtrak to Seattle, and flight back home with AS the dates were based around what redemption flights were available.