There's a lot more to it than mere military matters, Parliament and the general public were divided on the issues and there was strong feeling within the UK against the war and for US independence - especially amongst Whigs such as Edmund Burke, there was also the sheer difficulty of transporting troops and equipment across the Atlantic. It was not fought until the bitter end, the UK government conceded defeat earlier than it may have done otherwise, partially because it was expensive and unpopular at home. When it came to the War of 1812 a few decades later the result was essentially a US defeat and a Canadian and British victory (I think if you try to invade another country and you get kicked out and your capitol gets burned down that's a defeat - I know some nationalistic US history texts books say different but they are wrong) and the reasons for this difference have a lot to do with the fact that the justification for preserving Canada as part of the Empire was a lot more popular in the UK, so the government could commit much more to the conflict.

And yes the UK fought plenty of wars during the period between 1775 and 1783

First Anglo-Maratha War in India

Anglo-Spanish war around the Caribbean and within Europe

Fourth Anglo Dutch war

Second Anglo-Maratha War in Europe, India and the Caribbean

The fronts in Europe and Asia took precedence, for basic economic and security reasons.

There may have been a different outcome if the UK was only fighting on one front or if the war had been popular enough at home to allow the politicians to spend more blood and treasure, but as it was they reached a diplomatic settlement pretty quickly to get it out of the way to concentrate on the new threats caused by Spanish and Dutch ambitions and instability in France and to be able to commit to the conquering of the Indian kingdoms. India at the time having been a far richer prize than the US before the Western expansion revealed all the farmland of the Plains and Californian Gold and what have you.