most likely the petcock...
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1251932429/1

and if you haven't read this already...
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1344399573

(Sorry for the repeat of what Verslagen posted....we were evidently typing at the same time and he hit the send button first).

Do you still have the stock vacuum operated petcock - the diaphragms go bad and they either get stiff and fail to allow fuel to flow....or they get holes in them and flow fuel down the vacuum line.  It sounds like your likely is not flowing enough fuel - try turning the petcock to PRI which will bypass the diaphragm.

The kickstand safety switch can also be a problem.  This will kill the spark circuit if the circuit thinks your sidestand is down while the bike is in gear.  The test is to see if the bike runs when in neutral....if it stalls when you try to put it in gear, then the problem is likely the sidestand safety switch.  You can bypass the switch by going to the connector that is on the left side, just above the engine pulley cover, it is fastened to the frame with a plastic tie.  Disconnect the connector, and run a jumper wire between the terminals on the harness that is on the top (the lower harness goes to the kickstand switch).  If the bike runs well with the jumper wire installed.....it proves the problem is in the switch.

It’s a 1988 with 8300 miles

And if the carb has flooded and you just let it run you had better check your oil level ,  it could have flooded your crank case ,and gas is a lousy lubricant .Wrench first/ride later ,works best.

From the looks of those hoses on the vents.....somebody has been fooling around with that carb before you got to it.

Here are stock carb specs:

http://suzukisavage.com/cgi-bin/YaBB.pl?board=tech;action=display;num=1098869040

You most likely don't need a rebuild kit - just a good cleaning to get the crud out.  Most rebuild kits don't likely have the proper jets anyhow.

I would suggest starting with a #150 main, and a #52.5 pilot jet if you are near sea level....over 2,000 feet a #50 pilot.  Then replace the white spacer on the needle with 3 washers (size #3 machine screw - get steel as brass often has too big of an outside diameter).  If you end up having a slight lean surge at very low throttle settings - remove 1 steel washer and try running just 2 washers.

Can you show me a photo of the other side of the carb?  I want to see if somebody took out the brass plug over the mixture screw......and see if there is a port for the charcoal canister on the California model carb.

When I took the carb off last night there was about 5oz of gas in it. I ended up pouring it all out and it’s sitting on my counter drying out

Don’t reuse the carb bowl screws. Replace them with cap screws.