Looks like you did a few things wrong here, so I'll go over what I think may have been the contributing problems.
First off, I am guessing that you did not shift from 5th (twice) to 3rd as others here took your statement, but that you meant that you were in 5th, began braking and downshifting...and ended up in 3rd , meaning for at some point you were in fact riding in 4th.
What concerned me was your statement- "Upon dropping into 3rd, the rear kicked out, and the bike started to slide just as I was making the turn."
This sounds like you either downshifted at the begining of the turn, or a nano second before you initiated the turn...both not a good idea under any circumstances.
All braking and downshifting should ideally be done before a turn, and as mentioned you then should "power" through the turn keeping the throttle on and following a smooth arc.
Braking or downshifting should never be done while actualy in a turn,if at all possible.
Adding to this you mention that it just started to rain. This is the worst time to be on the road , because when it just begins to rain, the water tends to sit on top of the pavement and it's as slippery as ice on teflon. After it rains a for a bit, the water washes off the grime and oil left by cars, and gets absorbed into the road surface some, creating a better "bond" with any water sitting on top of it.
So the jist here is you should have slowed way the hell down to ride on this, and as they say "not make any sudden movements".
The Vulcan is not a sport bike, and partly due to the shaft drive, but mostly due to the fact that it has alot of engine mass moving , it is very easy to downshift and break the rear tire loose. Many times I have simply downshifted too aggressively (read that as "too soon") and "chirped" the rear tire. Not a problem while going straight on a dry road, but a real bad thing while turning on wet pavement.
As for the comment of you being in 5th gear at 55mph...sorry, but running at higher revs while on wet roads is not what you want to do anyway....and you can safely go 40mph in 5th gear on the vulcan. Most large Vtwins have alot of torque at lower engine speeds, and are quite happy thumping along in higher gears at low engine speeds. The thing of importance here is if you grab alot of throttle while doing this, the bike will take awhile to respond..which is exactly what you want when the roads are slick..no sudden movements. If you were in a lower gear, you would accellerate quicker..and thus risk breaking the tire free.
I am also thinking that although you did "brake and downshift" that you were still going too fast for the turn. On wet roads , making left and right turns through intersections usualy has me going slow enough that I am NOT countersteering throught the turn, but actualy driving through it like a car, with very little lean. Add the fact that you may have to ride over painted sections of road , that oil leaks from cars/trucks are at their worst at intersections and that most accidents happen while making a left turn or and oncoming car is turning left...then you'd almost think you'd be safer getting off the bike and pushing it ...so going slow becomes a question of "how slow can you go?"
I'm glad you managed to ride through it all and not get hurt or spill...shows you have the chops on riding buired in your head, you just need to brush up some. Go out to a empty parking lot and practice your downshifts...mostly so you can learn the highest...and lowest...engine speeds and to get a feel how much each effects braking.