No you cannot.
It IS legal to do so, yes. HOWEVER, neither FEDEX nor UPS will do it (by company policy) and you cannot, by Federal regulation, use the US Post Office.
The absolute easiest way to transfer the gun (Sorry, now I see you are in Texas, FTF is legal) is to meet face to face.
The next easiest way to transfer the gun is to have a local FFL mail the gun to an FFL local to the buyer. You both pay transfer fees, but then the gun can go by US Post Office saving quite a bit in shipping cost.
The last way to do the transfer is for you to ship, via FEDEX or UPS Overnight (again company policy) the gun to an FFL local to the purchaser.
You CANNOT, by Federal law, have a local FFL ship or mail the gun directly to the purchaser - that is only allowed if the gun (or a suitable replacement) is being returned to the original owner after repair or modification.
And, BTW, this part of the FAQ:
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In addition, Federal law requires that the carrier be notified that the shipment contains a firearm
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is FALSE concerning shipments within the same state or shipments going to licensed dealers/manufacturers out of state.