Hi, Quartney. Welcome to Flyertalk.
What often happens on these is that AA will only check what they consider reasonable connections. Further, to get to HNL in one day, you have to start out pretty early in the day from CAE or CHS, and if there is no availability on that flight, then it shows no availability for that day.
Perhaps an example will demonstrate. If I plug in CAE to DFW for June 12 (One-way, award), it offers me two flights, arriving at 12:50 P.M. and 7:30 P.M. There is an early morning flight, arriving at 7:45 A.M., but no award availability on that. If I plug in DFW to LAX for June 12, I get departures with MilesAAver availability from DFW of 7:35 AM and 10:05 AM, and you can't leave South Carolina in time to get either of those.
What to do? Well, you can leave CAE June 11 at 5:50 P.M., arriving in DFW at 7:30 P.M., and on June 12 leave DFW at 11:15A.M. arriving Honolulu 2:20 P.M. (nonstop). AA.com will never show you this, because it is a 15 hour 45 minute connection, but you can see T availability on AA.com by breaking the flights down into segments and checking as one-way awards. You would have to book this on the phone, but it could be done as one award, you would not need two. When you call in, have the flight numbers, dates, times, departure and arrival locations written down and just ask for those flights. Coming back can be done the same way. Leave HNL 5:45 P.M. June 21 nonstop to DFW, arrive 6:10 A.M. June 22 (flight 8), leave DFW 7:30 A.M., arrive CAE 10:55 A.M. (AA 3411). The problem on June 12th is what keeps the return from showing, if you check for a one-way on June 21 you will be offered those flights. Also on June 22.
So you can do the whole thing on the dates you want, but you have to leave the evening before and overnight in DFW instead of getting up at the crack of dawn to leave South Carolina. In DFW you would have two choices - Priceline a hotel at DFW (probably about $50) or sleep overnight in the terminal.