Just take KTX from Incheon Airport to Busan.
I agree. My wife just did this. She has the Korean retiree card so the cost was about 1/2 the regular fare (unfortunately, you have to be Korean to get this card, so I - and I assume you - won't qualify). The KTX (other than a few labor problems which haven't affected passengers) is safe, easy, and cheap. The KTX arrives at Busan Station, which is more convenient than the airport, and you don't have to wait as long for the KTX as for a connecting flight.
To mikesaidyes, I see you live in Kyonggido; I'm at Uijeongbu-si.