Leery is an adjective that describes something or someone as not trusting something or being suspicious of a place or person. It is usually used with the preposition of (e.g., I am leery of warehouses).
The noun form is leeriness.
Some American dictionaries list leary as a variant spelling of leery. However, it is more commonly seen as a surname and is considered by most to be a misspelling, not an alternative spelling.
Interestingly, the distinction between word and name is the same for the words leer and lear.
Leer is a verb to stare at someone in a wicked or perverted way.
Lear is the name of a famous English king immortalized in the Shakespeare play King Lear.
They are leery of giving any impression that their policies are not cast-iron guarantees, but opening bids for a negotiation. [The Guardian]
Bush Fatigue is real, as is voter leeriness about dynasties, but it’s not as bad in the primaries as it would be in a general election. [The Washington Post]
Paul Leary has filed a complaint with the Victorian Civil and Administrative Tribunal’s human rights division, on the grounds that the ban discriminates against him on the basis of his political belief. [Sydney Morning Herald]
A village hairdresser, who did not want to be named, said customers spoke of soldiers leering at young women jogging in the village; being aggressive to shop staff when they were refused alcohol and hiding from guards in bushes. [BBC News]