Such a coin does not exist. George III was born in 1738 and was king from 1760 to 1820.
That question doesn't have enough information for an answer, because it doesn't include the coin's denomination. A penny has a different value than a shilling.
You have described every British coin issued in 1797. Please provide the type of metal it appears to be made from, the diameter of the coin, the inscriptions and the design on the reverse.
Georgius iii
A coin marked with "GEORGIUS" or "GEORGIVS" could be any British, British Colonial or British Empire/Commonwealth coin issued during the respective reigns of any of the six King Georges. George (I)(1714-1727) George II (1727-1760) George III (1760-1820) George IV (1820-1830) George V (1910-1936) George VI (1936-1952)
No. King George III died in 1820. King Edward VII was on the throne in 1907. All 1907 British and British Empire/Commonwealth coins will have King Edward VII on the obverse. You may have a token or medallion in remembrance George III.
Your coin is either a British 1797 "Cartwheel" Twopence 41mm in diameter, or a British 1797 "Cartwheel" Penny 36mm in diameter. If somebody has drilled a hole in it, it might be worth a few Pence. Modified coins have no collector value.
Try English. The Brits call them Pounds and the One Pound coin was first issued in 1983. The Sovereign, which is equal to One Pound was reintroduced into the British currency in 1817. George III died in 1820, George IV was on the throne in 1821.
Your coin could be a gold Guinea (with a crowned shield) or Half-Guinea (with a crowned "spade" shaped shield) or a copper Halfpenny or Farthing (with a seated Britannia), the only four British George III coins minted by the Royal Mint in 1773. Most coins minted during this period had the reverse upside down to the obverse and had no denomination on them. Can you provide more information?
50 dollars
You are asking about a British coin dated 1788 ("Georgius 111" refers to King George the Third). You would need to give a better description of the coin (condition and denomination, or at least a physical description of the coin's size, material, design, etc.) to get any kind of an estimation of value - it could be less than a dollar or could be several thousand. Also, there were many tokens and counterfeits - mostly copper or brass - made around that time period, so your "coin" may fall into one of those categories as well.
It depends how worn and corroded it is. It will be either a penny (weighing 1 ounce) or a twopence (2 ounces). For more information see the values pages at the related link.