The refrigerator side is warm

maybe the fan in the freezer that circulats the air flow to the frig side has failed

Yes, if your evap is fully frosted, and the fan is not working. The frosted evap will keep it cold but since the fan cant move the cold air from the evap it'll be cold, but not as cold as you'd like.
http://parts.sears.com/partsdirect/part-model/Frigidaire-Parts/Refrigerator-Parts/Model-FRS6HR4HW2/1428/0165000/R0607119/00005?shdMod=FRS6HR4HW2
there is the screen you need to order a new fan motor. option number 13 is the fan motor. lmk if u need more help!

Contact a professional. Sounds like either your damper door is not operating correctly and not letting enough cool air from the freezer enter, or there is an air leak that is making ice form .

Replace Defrost thermostat which is fixed with evaporator coil.Sure your problem will be solved.

Freezer fan motor is probably not working properly. Cold air is not being transferred from the freezer to the refrigerator.

you may have defrost problem check see if you have frost build up in the freezer back wall

If you have frost on the back wall of the freezer compartment you have a defrost problem. When the freezer coil does not defrost the refrigerator does not get air for cooling.

take all the bottom shelves and baskets out of the freezer. look at the back wall of the freezer. youll see a block of ice in a metal pan looking part at the bottom of the rear panel. there is a hole in the bottom of that pan. thats the defrost drain. its clogged. use a hair dryer to get all the ice out then use hot water to flush out the drain hole.

If the frost build up is on the back panel of the freezer then it has a defrost problem which causes a no cool situation. Let me know more about this frost buildup and what you have done.