More than one way to skin a cat.
I have a cooler with a raised bottom. Eels on top / ice under. Keep eels out of the water, but keep them cold. They appear to be sleeping and move very sluggishly.
I'll hook one of the cooled eels under the chin and out through the eye and cast it out. No finder, sinker or whatnot generally.
VERY rare to have a "monkey's fist" this way. At least from my experiences.
The warmer water brings the eel back to full life force. Never let the eel sit on the beach. If you let them hang between cast, they'll curl a bit, but never had one ball up.
As to where to cast?
Read the beach. Find where the wave action comes together from opposing sides. Where they meet, the wash will be taking out most of the time if positioned right. (may need to go one side or the other a bit though)
Heave it out and let it sit a spell. How long? Depends on you and how the eel travels in the drink. Sometimes 20 min. to a half hour. Other times, 5 minutes.
If the eel swims out of the little rip, the tide will usually push it back to shore. Catch it right and it heads for England.
The above is for open beaches. Not rocky areas or weeded jetties.
I hate to throw an eel away just because it died! At $2.00 each, they get wasteful.
Do a Google on rigging an eel. Needles are easily fashioned from wire coat hangers or (better yet) stainless steel welding wires.
Frozen rigged eels last a very long time in the freezer. Wrap individually in wet cloth or paper towels.
If time is tight, Simply skin and salt (google again) and add to a plug for eel skins. Inside out or right side out both work good.
The convenience is well worth the effort to save them.
Just some more ideas for your eeling pleasures.
I will always chill my eels no matter where I intend to fish. Much
easier handling and keeping that way.