[hatzie]

The high speed blower relay is kicking over so the power is being routed from the J studs through the relay and directly to the blower
This means power in to the switch is OK and the high speed wire from the switch to the relay coil is OK.

Check your Low and Medium control wires from the switch to the blower resistor with a headlamp bulb when they are selected.
If it lights up bright then the resistor is screwed up.
If it's dim or doesn't light up then the switch or wires or both are not happy.

I use a 9006 headlamp bulb in a spray paint lid with a connector pigtail I liberated from a vehicle headed to the crusher. Kinda Ghetto but it works. 55Watts should be enough power draw to test this circuit.
A meter can read 12v or 14v when the switch or wiring is dodgy since it doesn't pull a lot of current. A test lamp only has a tiny little low wattage bulb so again it will not pull enough current to tell you a lot. A headlamp bulb can stress a dodgy switch connector or wire.

IIRC the switch terminals are Pak Con II and the resistor terminals are Packard 56. The connector part numbers are in the wiring diagrams so you can look em up to find out what family they are.