I don't think you'll have any problem finding something nice in your price range. I just checked out The Sun News Rentals:
You can check more than one option under Type of Housing. I entered Max. rental 1000 and clicked Durnished, simgle family, corporate short term, single family home, and cacation/seasonal.
I'm not familiar with all the areas of the Grand Strand, but in this list I noticed some that are considered good locations such as Carolina Forest, Palmetto Point, Coquina Harbor, Myrtlewood, Barefoot Resort, Blackmoore Golf Club. I only looked at the first two pages. If you come across something you like, post a message with it's location and I or someone else on the board will check it out for you.
Also, check out Vacation Rentals by Owners (VRBO.) These are individual owners who may be willing to cut you a better deal than rental companies.
Best wishes in your search. I hope you find something nice to rent as you begin to look for your new home at the beach!