FD750D Carb Issues

I have a 27HP Kaw FD750D on my Scag Turf Tiger, and my carb is in need of a rebuild, or worst case replacement. Problem is the engine ID sticker is no longer on my machine. Is there any other identifying marks I can go off of? I can't seem to find a rebuild kit, and $400+ for a carb is hard pill to swallow. If I have to buy new, I will do so, just need to make 100% sure I am buying the right carb.  I am also looking for a replacement throttle linkage (thanks to the previous owner).  
Scag Info: 
Model: STT61A-27KA
Serial: 8521775
Any help and/or insight is greatly appreciated. 
 
Thanks, 
Fish 

SRTsFZ6, thank you for pointing that out! Very much appreciated. I am trying to figure out what all the different jets mean in the exploded view. Assuming they're model specific. 

At first it was backfiring, losing power,  & missing bad so, I replaced both coil packs(set my gap in accordance to KAW) & spark plugs (gapped per the manual as well). Fired back up, and the miss was gone, and the backfiring became more intermittent. So, I did a compression test (passed), and decided to remove my carb to clean it.  Thoroughly cleaned the carb, all surfaces, and reinstalled. Backfiring was gone... for now.  Mowed all day, no backfiring but, I had increased fuel consumption. Mowed a second time, no backfiring, increased fuel consumption. 3/4 of the way through my second cut, I stopped to re-fuel. After restarting, the backfire was back. I run only ethanol free fuel, and have a secondary filter before my fuel pump. Fuel pump & solenoid are both newish (replaced last season). All of this lead me to believe my carb is in need of an overhaul.  I just ordered all the gaskets, floats, jets, etc. 


 
^This matches what I have on my engine 

I'll try the choke, and I the solenoid under the carb is new too.