I'm a big fan of cleaning fish quickly after they are caught and then taking good care of the cleaned fillets. Wash them thoroughly removing excess blood and then keep them as cool as possible until it's time to eat them.
What I think affects fish texture and flavor is towing fish around on a stringer for any length of time beyond 20-30 minutes. Stressed fish release chemicals from their organs that are eventually absorbed by their muscle tissue, the part you eat. If the water is warm say over 60 degrees that speeds up the process significantly. And if they die on the stringer those fish are certainly going to be low quality fare.
For keeping fish cool put the clean fillets in a plastic bag and then into a canvas or other cotton bag that is kept wet and always in the shade. Fish will be ok for several hours kept this way. I like to hang fish in a shady spot where there is a little breeze which enhances evaporation and cooling. They will also dry out more quickly in the breeze so keep an eye on them.