Your air conditioner is a dehumidifier. In Minnesota we got dew points well above the indoor temperature but as long as the air conditioner was used to bring the house down to a reasonable temperature, the humidity was plenty low. I managed to pull humidity down to 25% during the hottest weeks of the year and most weeks I'd have it in the low to mid 30's. If I let it get warmer, aka less use of the air conditioner I would have humidity in the 40's. I aimed to keep the dew point of the house below 50 degrees.
The success of an air conditioner dehumidifying your house is dependent on long cycle times, which becomes difficult when an HVAC contractor oversizes the A/C. If the blower speed is too high and the coil isn't cold enough to dehumidify the air, you have the same problem. Also if you don't have a well sealed house, you are letting humidity in, which seems to be the most likely scenario for you if you are having this problem but it is probably a combination of those issues.
The issue with dehumidifiers is that they essentially are an air conditioner with both the indoor and outdoor components inside, they leave you with a net gain of heat while drawing off moisture to a catch bucket or a drain. I made the mistake last year after reading on the hvac-talk forums that dehumidifiers were more efficient at removing humidity so I monitored my humidity and kept it at 40% but the dehum. never shut off to keep it there and burned 400watts to no end. The house was warm but the humidity level kept it comfortable enough. I ended up with the highest electric bill. I dehumidified my house using the air conditioner the following month when it was hotter and my electric bill was actually lower.
In the summer if you have issues with your humidity being that high, you should try letting the house get a little bit warm and try running your air conditioner in one cycle 4-8 hours long and tell me what happens to your humidity level. Running a long cycle will keep the a/c running its most efficient and remove the most humidity. These 30 minute cycles that some people have get rid of very little humidity.
This summer I'll be trying to use a 10.7 EER(energy star) 5k BTU/hr window A/C which is a higher EER than my current air conditioner's SEER(1986 unit) and will be in my bedroom where I want the most comfort, mostly for sleeping. If it can't keep up cooling the whole house, I will close the bedroom door and when it is overcooling the bedroom I'll open the door. If needed I'll cycle the furnace blower for a bit to suck the air around or use a few of my box fans. I got a good knack for working the 23k whole house unit last year to keep the humidity down and expect this unit if setup and used properly will do better and be cheaper.
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