Coconut Battery is a marginally useful App at best IMO, because it cannot account for the multitude of factors that cause lithium cells to wear. While cycle count is one major metric used in estimating service life, there are other factors that can cause a lithium battery to go kaput long before reaching the end of that expected cycle count.
I agree it sounds like the battery. An Apple Authorized Service Provider can perform MRI on your system (it's free) - be sure to take your charging cords to have them tested as well as problematic chargers or charging cables can sometimes play into battery issues. Chances are the MRI graph will show your battery in the red and everything else as OK.
For good measures, it's not a bad idea to periodically visually inspect the battery for unusual smells, physical deformities, swelling, or leakage, if you havent already.
