That model doesn't contain a relay.
When you say "raise the cores all the way up" do you mean that the top of the tuning slug is DEAD EVEN with the rim of the hole in the can?
If so, I can offer a suggestion.
Looks a little like this.
The solder pads under L8 and L9 reveal that one side of each coil has two pins. The opposite side of each has three pins. Take a 30pf (approx) trimmer cap and solder the two leads to the outer two pads of the three-pad side of each coil.
Set the slug in the coil two or three turns clockwise from that "even with the rim" position and then try to peak the trimmer cap for each coil. There's some back-and-forth that may be needed. If the trimmer cap is going all the way to either max or minimum setting, the coil position gets tweaked to bring the cap to a position between those extremes.
Not a very complete procedure, but your other choice is to replace L8 and L9.
Unless there are additional faults in the receiver section, this should restore the receiver's sensitivity.
73