Sure looks like a slot issue and you're right if it is the slot(s) then your looking at the logic board to be replaced and that will run you somewhere in the $700 range,
If you are near an Apple store I would take the system and have them diagnose it. there will nor no charge for that even being out of warranty, and see what they say. If it is the logic board you can at least make an informed decision as to if you want to spend the money.
If you don't do the repair there is one possibility to make the system more useable, get two 4gb sticks. At least then you'd have 8gb of memory and only be out the cost of the memory.