"is labeled as 240v but it's actually 2 phases giving
208v"

That is about a 16% drop in voltage, it should not cause smoke.
Question # 1 becomes, why did the old one go bad ? You say the
transformer is overkill on the current (you don't do that with
voltage) so whatever short that may have blown the fuse or thermal
fuse in the transformer may still exist. That's if it is a short.

If it is not a short, is it possible you screwed up on the primary
wiring ? Let's put it this way to just cut the trees down now - did
you actually measure the voltage at the secondary before applying it
to the circuit ?

Deal is this : If the voltage is correct you have a short. If the
voltage is 184% or double what it should be, you have a short NOW.
Some components fail very quickly with too much voltage applied and
others do not.

"The output isn't even connected to anything unless you're pulling
the
trigger but the smoke kept getting worse even without the saw
running......"

I think it very possible you got the primary wiring wrong somehow, or
maybe that it was the wrong transformer. Luckily there is not alot of
silicon around but DO look for anything that looks like a diode. It
may be a Zener intended to protect the trigger switch from arcing.
These things usually short when subjected to excessive voltage so
therefore, it might be energized all the time, not working, and
releasing magic smoke whenever powered up. It makes sense.

Get in there with an ohmmeter, DVM or whatever. Those cheapo tool
places like Harbor Freight have meters for about six bucks. They are
not quite a Fluke but they are good enough for most everyday tasks so
no excuse. If you disconnect the secondary of the transformer and it
still smokes it is bad. You may have made it bad with too much
voltage, BUT, you could have gotten a mismarked part, or the specsheet/
data on it was faulty or for some other part.

If you are wondering about the difference in a two phase and three
phase, 208 is less than 240. That can't fry much. BUT -

Ummmm, you didn't connect the neutral by chance did you ? You don't do
that, it simply doesn't work. There is a possibility that if you
hooked up the neutral, with two legs feeding primaries, you only blew
the transformer. You can't do that unless the phases are 180 out, not
120 or anything else. If that's what you did, get another transformer
and leave the white wire alone :-)

J