The service information for this vehicle has conflicting information; apparently there was a transition from the older relay based design to the newer BCM controlled design with a power module in this model year. The diagnostic information and parts locator information both mention a blower relay located on the shroud, but the electrical schematic indicates a solid state power module that takes the place of and performs the functions of the mechanical relay. 1994 model service information does not list a relay at all, so '93 must have been tre transition year.
Images available only to Customers.
As you can see from the electrical schematic above, the blower motor power is controlled by a blower power module (electronic relay) on the newer design. The power module varies voltage provided to the blower motor in response to commands from the vehicle's body control module. If the blower is inoperative, the vehicle could have either a faulty blower power module, a faulty body control module, or a faulty control head (climate control assembly).
The erratic operation you described the vehicle having initially (blower not shutting off when the key is in the off position) would indicate that the power module (electronic relay) is probably the fault; when the key is turned off it powers down the control head and BCM, although the power module has a direct battery connection at the fuse block.
It may interest you to know that the module is available from AC-Delco (part #15-8548) for a retail price of 147.11 if this is helpful.