It is almost certainly a mat of hair and soap scum caught on the crossbars a few inches below the strainer, not a blockage deep in the pipe. Pull the strainer and hook it out with a barbed drain tool — three dollars, five minutes, no chemicals.
Look in the obvious place first
Nearly every slow shower drain in a house of any age is the same thing: hair binds with soap scum and body oils into a felt-like mat, and that mat catches on the crossbars sitting two to four inches under the strainer. It is close enough to reach and it is almost never further down than that.
Unscrew or lever out the strainer, then feed in a barbed plastic drain tool — the flat toothed strip sold in every hardware store for a couple of dollars. Push it in, twist, pull it back slowly. What comes out is unpleasant and it is also your answer. Run hot water for two minutes afterwards.
Skip the chemical drain cleaner
It rarely clears a hair mat, because the mat is mostly keratin and the caustic products in the bottle are formulated for grease. What it does reliably is sit in the trap, generate heat, and attack older PVC joints and the chrome on your strainer. If it does not work you have also now filled the pipe with a caustic liquid that the next person to open it has to deal with.
If the strainer was clear
Then the interesting question is whether it drains slowly or drains and then gurgles. A gurgle, or a glugging noise from a nearby toilet when the shower runs, is a venting problem rather than a blockage. The drain is struggling to pull air and is stealing it from the nearest trap — which is also how you end up with the sewer smell we covered in a previous post.
The Florida-specific one
Older Pinellas houses can still have cast-iron drain lines. Cast iron scales up internally over fifty or sixty years until the effective bore is half what it was, and a shower on a scaled line drains slowly on a good day and backs up on a bad one. A camera inspection settles it in twenty minutes. If that is what you have, no amount of pulling hair out of the strainer changes anything.
When to call someone
More than one fixture slow at the same time, water backing up into the shower when the washing machine drains, or a drain that clears and re-blocks within a fortnight. Those are line problems, not strainer problems, and rodding a line is not a job worth improvising.