UKH

One who chases old playwright out of Florida (6)

The answer is ‘etcher’ - that’s the ‘one who chases’, but where do the playwright and Florida come in to it?

Cheers. 

https://en.wikipedia.org/wiki/John_Fletcher_(playwright)

Ah! I get it now. Wouldn’t have worked that out in a million years! Cheers. 

John Fletcher, old playwright.  Lose the FL which is state code for Florida to get etcher. 

My guess is the Florida reference gives you FL.  Add this to your 'etcher' and you get Fletcher.  Wikipedia tells me that John Fletcher was a Jacobean playwright.

Not so hard with perfect hindsight, but I doubt I'd have got there otherwise

Really clever clue, though ‘out of’ is a clumsy way of saying ‘without’.

jcm

I thought the recognised abbreviation for Florida was FLA. Has that changed?

Urchin: Got any gobstoppers, mister?

Shopkeeper: Sorry, sonny, you're without luck.

Bit difficult if your knowledge of the state code for Florida is from Lou Reed.

"These official state abbreviations have all been standardized by the United States Postal Service. State abbreviations are all two letters, and these two letters are always capitalized without any periods"

Crossword compilers seem to like that pattern, e.g.

"Composer who cuts short their dinner (5)"