The main definition of the adjective fleshly is of or relating to the body. It’s synonymous with carnal and sensual, and it’s often used in religion-related contexts as an antonym of spiritual or heavenly. Things that are fleshly are often regarded as sinful or at least separate from the spiritual side of life. Fleshy has nothing to do with spirituality. It means (1) relating to, consisting of, or resembling flesh, or (2) plump.
Many dictionaries list fleshy and fleshly as variants of each other, but the words are only rarely used interchangeably.
Examples
Fleshly
As death is antithetical to life, it should follow that fleshly pleasures would have no attraction for the soul. [Trinidad Express Newspaper]
Living a fleshly lifestyle of disobedience to the Lord clouds our eyes, diminishes our ability to hear, and fogs our thinking. [Christian Post]
A concurrent thread running through the early church in America was that of the Shakers – who shunned all fleshly indulgences, but had a passion for God that actually set them a-quiver. [Minnesota Public Radio]
Fleshy
It is not just consumers who seem keen on the fleshy fish. [Financial Times]
Olives were already on the table, and they were really good ones, flecked with herbs and fleshy on the inside. [Montreal Gazette]
Crotti, a big, fleshy man, is plainly enchanted by the delicate Chan. [Times Online]