Here is the DirecTV Phase III dish we think you are talking about.
http://www.solidsignal.com/prod_display.asp?main_cat=02&CAT=&PROD=OD1820-BX
The electronics built into the LNB assembly will get the three DirecT birds, and will feed four receivers for subscription use.
Now, to get
either Dish Network 119º NASA or 110º GolTV, that dish will need to be re-aimed.
That is just an inescapable fact.
It's not hard.
The dishes are not that directional, and the signals are very strong.
I do seem to recall using a very old junker Dish receiver , well past it's prime, and still being able to get NASA on it.
That was back when Dish used
BLUE activation cards.
Since they went to the yellow cards, maybe 3 years ago, the cards need to be periodically re-activated (so to speak) to keep them working.
So, a yellow card or receiver that had been stored in a closet for a few months, might not get anything.
Whether or not it'd still get NASA under those circumstances, someone else will have to make that specific claim.
Yes, any FTA receiver will get GOL and NASA just fine.
You can get a cheap Traxis 3500 for about $70, and I think I've seen a Satpros 500 on Ebay for $50.
Of course, what you wanted was a free solution, and all we've given you is expensive and awkward.

Sorry 'bout that.
If anyone has a better solution, do speak up!