You have got a lot of advice, some very good. This must be confusing to a homeowner and the local tech.
I see the priority this way.
1. You need a 45-50^F a/c cooling coil to get to <50%RH at 75^F . This is more than a 25^F temperture drop of the air flowing through the coil to remove the right amount of moisture quickly.
2. The a/c needs a cooling cycle long enough to load the cooling coil with moisture and drip to the drain. 10 mins. is too short. It takes +20 mins. to load a coil and start draining. Slowing the fan will get you the cold coil. Increasing the dead band of the t-stat to increase the time of the cooling cycle. During the hottest days, your a/c will maintain <50%RH. During days of low/no cooling loads, it is impossible for any a/c to maintain <50%RH when the outdoor dew points are +55^F. You need supplemental dehumidification to maintain <50%RH which is a must to avoid mold and dust mites. On cool wet days, you must dehumidify 3-4 gallons of moisture everyday to maintain <50%RH. This is the moisture from the occupants and the infiltrating/ventilating fresh air.
You have additional indoor air quality problems in addition to high indoor %RH. You need an fresh air change in 4-5 hours to purge indoor polutants and renew oxygen. This would be 50-70 cfm of fresh air when the home is occupied at a minimum.
I suggest you get you a/c setup properly and then get a small whole house ventilating dehumidifier like the Ultra-Aire 70H. This type unit will provide the right amount of friltered fresh air on a occupancy schedule and maintain <50%RH when the a/c does not have enough cooling load.
Fresh air ventilation and <50%RH will make your home healthy and and comfortable. It will take a month or two of fresh air and <50%RH to stop the dust mites and mold from affecting your allergies.