Passenger side

it is either a heater core gone bad or if A/C the drain is plugged. so check both if hoses go inside to connect to heater could be that as well.

So if it's heater core then what? How do I check if the A/C drain is plugged?
Should I peel the carpet on the floor back because of all the water that's been soaked down there?

for a plugged drain only if you have a/c otherwise there won't be one, go under car and look near center of car for a small hose, run a piece of hanger up it and prepare to get wet. for getting water out a shop vac is good it will suck out most of the water just take the paper filter out so it won't get destroyed. there may be a rubber plug under the front and back take it out to drain also then put it back in but there always isn't one