My 97 Jetta GT is having problems.  Backfiring when accellerating or under a
load (uphill), lacking power at lower rpms, and while this is happening it
makes a sort of putt-putt sound that sounds like it is coming from
under/behind the front passenger seat.  Much worse in the rain (this tells
me "electrical" but when I look under the hood everything seems dry).

I did some basic stuff.  Checked the wires, changed the oil and air filter.
The air filter was gross, covered with crap blown back from the engine; I'd
never seen an air filter dirtier on the engine side than the intake side.
Threw in a bottle of dry gas.  Helped a little.  Not enough.

Took it to a local garage.  They did a pressure test - good.  Checked plugs
and wires (which were only a few months old) - good.  Checked the MAF -
good.  They decided I needed a distributor.  To their credit, they told me
their supplier wanted $300+ for one and that I could probably find one
cheaper, which I did.  They put it in yesterday and I picked it up today.
Problem not solved.  May be even worse, though I think that's probably
psychological.

I don't want to take it back to a garage without more information and
without trying things myself.  I tried disconnecting a couple sensors to see
how the engine reacted.  Looking down on the engine you have the four spark
plugs (I'll call them P1 - P4) and then a sensor just in front of them to
either side (I'll call them S1 & S2).  They're arranged like this:

P1  P2  P3  P4
 S1           S2

What are the two sensors?  They look identical but might serve different
purposes.
When S1 was disconnected the engine ran very rough.  When S2 was
disconnected it didn't make any difference.
Is this normal?  Does this mean S2 is bad?

And if so, how does this relate to the symptoms?  Why, for instance, is the
problem much worse in the rain?
Is there anything underneath that's electrical I should check?

Any advice would be welcome.  Thanks.

Jim