The oil leak could be one of the things your mechanic listed but I really doubt it’s all three. If it started right after an oil change, you could have a leaking oil filter gasket, or worse it could have been double gasketed. I say worse because if double gasketed, it could suddenly and catastrophically fail and you could lose all your oil and possibly the engine if you don’t shut down quick enough.
The oil pressure switch would be a second choice if it got damaged at the last oil change. They are usually located right next to the oil filter and can be hit by the oil filter wrench when removing the oil filter. They usually dump oil a lot faster than you are experiencing though.
Both of those are not very expensive to replace. I assume by oil filter housing, he really means the oil filter mount, which I doubt would leak.
Oil pan gasket don’t usually start leaking all of a sudden like that. They generally start leaking very slowly at first and gradually get worse over time.
The cheapest way to fix this might be to just get another oil change and see if that doesn’t solve the problem. Use a good mechanic, not a quick oil change place.
And you might consider an ATF exchange for your transmission. Fresh fluid can do wonders both for performance and adding life to the transmission.
Edit: for the ATF exchange, use a good mechanic or the Ford dealer to insure that you get the correct ATF and not some generic that may or may not work. The dealership would be your safest bet on this even though it may cost a little more. In the long run, it isn’t that expensive for the potential payback in transmission life extension. Do not use a national chain transmission shop.