Unless the unit is in defrost, you'll likely need a new compressor to solve this problem...up to you if you want to invest $400 or more into it. Find the defrost timer and advance it to see if the compressor kicks in or not. Hopefully you wired everything correctly when you replaced the relay/overload. There's always the outside chance the sealed system has a refrigerant leak, in which case throwing money into it is futile.
The "Kenmore" you have is actually a Frigidaire.