Use the bucket system to transport to the beach in the truck and an eel bag clipped to your wader belt for when your working the beach on foot. Keeps your hands free. Also, clip a small washcloth to the belt as well to deal with them. Aquaskins makes a nice mesh eel bag. The bag is strong mesh, keeps them cool and they get rinsed from time to time while fishing.
When I go back to the truck to head to another spot, I simply clip off the bag and clip it to my cooler (outside) or drop it in the bucket. IMO six eels will last you plenty of time while you are on foot between spots...probably all night. If bluefish are cleaning you out...switch to plugs or move.
A lot of ways work, I just like to keep my hands as free as possible and only like holding a rod. (and a big fish of course)
You just want to AVOID the bucket of eel slime (eel juice as my son calls it) spilling inside your wifes car...She will leave you. (take my advice on that) This smell will get VERY nasty the next day in the hot sun. I keep them in a bucket the the cooler on the front bumber now.
But...its worth the trouble to deal with them. They are a hearty bait and work great.