My Kitchenaid refrigerator deli drawer freezes everything, how can I fix this?

is the freezer freezing too? to much ice build up? if not check ur damper vent, make sure its not stuck closed or open to far, check the thermistors, the tempeture control thermostat, defrost timer and defrost terminator

Call in a repair man, it shouldn't cost that much. Yes, your food will be effected without that plastic door, as the drawer has controls and cold settings just for deli items. You need a special glue for cold. I'd call in a repair man, or order a new drawer replacement from the manufacturer.

Hello friend, thanks for your inquiry!

I understand that you have a Frigidaire refrigerator, FGHN2866PF1, and your deli drawer is freezing items. You may want to turn down the temperature within your deli drawer, when adjusting the temperature could take 24 hours for your refrigerator to stabilize. If that does not resolve your issue I would suggest removing the drawer and checking over to make sure there are no cracks where too much air may be coming in. If there is too much weight in the deli drawer that could cause the drawer to sag, which may release cold air into the deli drawer. I hope this information has been helpfu. -Matt

The temperature in the section or the refrigerator isn't set correctly.
Also, check the thermostats if it is functioning correctly. PLEZE LET ME KNOW.

There is a chance that your refrigerator needs defrosting. Remove everything and unplug your refrigerator, open the doors and let stand for 12 hours. Sometimes the air vents restrict with ice causing this trouble.
http://www.on-timeapplianceplano.com/refrigeratorrepair.html

I would direct you to this link to find the part number you're looking for. I do not have one of these units to look at so am unsure which specific piece you'll need to replace.
Model 59673503200 KENMORE ELITE KENMORE REFRIGERATOR Parts

have your thermostat setting in the right setting otherwise it will happen again and again. have someone to check it like authorize repair shop or technician

check the cold control or the door is not closing proper

First, is the temp really 40??  Get your outside thermometer and stick it in there.  You may get a surprise.  If so, see if the thermostat will respond to any changes.  It is possible that either the sensor or the electronics have gone bad...I'd vote for the sensor.  
the other possibility might be low gas...but I'll want to see what the temp in the unit is and whether the changes you make do anything.

this particualr refrig is notorious for this. check your controls ( your freezer control as well set to 4 max) any higher and the refrig is running longer, generating more cold then needed, behind the left hand crisper is a few slots( the fresh air return) by default the coldest spot in refrig, i usuallly would reccomend to my cust not to keep high water or thin walled vegies in there such as carrots cukes tom and lettuce, perfect beer and fresh /frozen defrosting meat to be though, if you ever looked at the newest bottom freezers they've pushed up the drawers and added a deli pantry, running the width of frig