How can I tell if slow drainage is caused by a problem in the main sewer line or the venting system?
I'm afraid it's probably both. You have a venting problem because the sinks gurgle. Watch the water level in a toilet when the washer starts to drain, it'll probably drop a little. The reason you don't have a bigger problem with water getting sucked out of traps is because the main line is obstructed, minimizing the demand for makeup air through the venting system. If the mainline were not obstructed, the backed up water would quickly drain away once the traps were voided of water, providing an alternate path for makeup air.

The fact that toilet paper is a contributing factor also indicates main line problems. The waste water backing up could have caused solids to float up into vent lines and get lodged there. Hopefully snaking through the roof vents will free any vent obstructions. Depending on the nature of the blockage, simple snaking may only temporarily allow drainage. More aggressive clearing may be called for.