Hello Arnono,
Well.... I have to tell you that August is a difficult month to find accommodation. You will certainly find but you won't have great choices. Usely, when the ferries arrives on the port of Naxos, there are at least 100 people there try to convince tourists to go to their hotels and rooms. But August still a dificult month. There is also an accommodation desk on the port.
Then the prices. The are getting really high in August even for medium accommodation. You should check
http://www.greeka.com/cyclades/naxos/island/naxos-last-minute.htm but even now, only one have made a post, as many of them are already full.
For cheap and confortable accommodation you should maybe try the rooms of the campings:
http://www.maragascamping.gr/ and
http://www.plakacamping.gr/
Good luck and have great holidays (I love Naxos will also be there for a small week in August)