Gone are the days when you used to get an advantage paying with foreign currency when Jamaica started to allow her citizens to readily access foreign currency in their own bank account. In the early days (in my case early 80s), everyday Jamaicans were told by their government that "changing money" from Jamaican to a foreign currency was "unpatriotic".
In fact, in those days the Jamaican Dollar was actually stronger than the US dollar but you needed US dollars to go to Miami to buy goods to bring back to sell to make a living. Hence, the "higgler" in auto parts etc. was born. As the Jamaican Dollar started to lose its value against the US dollar, speculators would offer more Jamaican Dollars than the actual value of the US Dollar (and I am sure other currencies but I don't have first hand knowledge about those) because it was not possible for them to get US dollars at a bank and it was illegal to take Jamaican Dollars out of Jamaica.
Now a guy who changes money outside a bank or cambio has no real reason to give you more for your foreign currency because he can get it himself anytime he wants it. His only angle now is to either shortchange you or to slip in a few counterfeit bills to make his money.
My advice is to just deal in Jamaican Dollars and get used to the exchange rate. It is true that tipping in US dollars lets the person who gets the foreign money save it for a later trip to the US but it also becomes a hassle for them to change it to Jamaican Money for everyday transactions.
If you go into a small business and pay for a $800 Jamaican meal with a $10 US bill, I would bet that you would not get any change in return and thus lose @ $.80 cents on the transaction. Multiply that by several purchases over several days and you are just throwing money away. In fact, when is the last time you got any coin currency back? Anything less than $50 Jamaican will usually just be rounded off anyways.
Everything is priced in Jamaican Dollars anyways (except at resorts or other fancy-dancy places) so just pay the price posted and be done with it. If it is a service you want, then pick a price you will pay and negotiate from there. It really isn't all that difficult it just takes practice and that means you will just have to come back to Jamaica more times to practice!
Peace and Guidance