as for your new battery, sounds like the connections are definitely not tight enough . . . based on the description and no pictures yet
As for the first battery, how old was it? It sounds like it was either pretty old and failing . . . and repeatedly jump starting a vehicle and driving it doesn’t really bring it up to a full state of charge . . . and you may also have a parasitic draw
When you do manage to upload the pictures, that will be a big help. thanks
anything getting hot or melting is due to a poor connection creating resistance. Everything needs to be clean and tight. If the switch is damaged and they cut off the original terminal to add this switch, it may be less hassle to simply replace the switch and make sure everything is clean and tight before charging the battery with a trickle charger before running it again.
If it is a switch like TwinTurbo described, I would recommend replacing the negative cable. The negative cable is fairly inexpensive. The positive cable should be cleaned and inspected. Yes, fully charge the new battery before attempting to start.
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Once you restore integrity to the wiring and get the battery charged, you’ll need to find out if the switch was added to compensate for a parasitic draw that was draining the battery.