I did this last week. arrived at DMK. went to the info booth to ask about the free shuttle to BKK. Information said you must have a connecting ticket to use the shuttle. so they told me take MRT bus 510 to BKK. I just missed it, while waiting saw 504 come with suvarnabhumi airport markings. got on 20(?) baht ~ 30 min,
Had to wait 3 hrs at BKK for a bus to Pattaya, very busy, buses all full. Pre-purchased my return to BKK ticket online.