Oldsmobile intrigue High beams will not turn off

The switch is bad >
Each headlamp is individually fused so that battery voltage is applied to the individual lamps at all times through circuit 1640 for the LH headlamps and through circuit 1740 for the RH headlamps. When the headlamp switch is in HEAD, the ground is applied directly to headlamp switch through circuit 1750. When the headlamp switch is placed in LOW, ground is applied to the low beam headlamps through circuit 1201. When the headlamp switch is placed in HIGH, ground is applied to the high beam lamps and to the high beam indicator through circuit 1200. The high beam indicator is fused so that battery voltage is applied at all times through circuit 640.

The box you hear the click in is the daytime running lights control module .
Do you know how to do electrical testing using a DVOM- digital volt ohm meter or a test light ?

Low and High Beams Headlamps Inoperative
Step
Action
Value(s)
Yes
No
1
Was the headlight system check performed?
--
YES - Go to Step 2
NO - Go to Headlight System Check
2
Does flash to pass function normally in the headlight system check?
--
YES - Go to Step 3
NO - Go to Step 7
3
Does the autolamps low and high beam operate normally?
--
YES - Go to Step 4
NO - Go to Step 6
4
Turn the headlamp switch to the Head position and the headlamp dimmer switch to Low position.
Using a test lamp, backprobe between connector C201 terminal W and ground.
Is the test lamp on?
--
YES - Go to Step 5
NO - Go to Step 6
5
Repair poor connection or open in circuit 1750 between connector C201 terminal W and S283.
Is the repair complete?
--
Go to Headlight System Check
--
6
Replace the headlamp switch. Refer to Multifunction Turn Signal Lever Replacement - On Vehicle for service procedure.
Is the repair complete?
--
Go to Headlight System Check
7
Repair poor connection or open in circuit 1750 between S283 and ground.
Is the repair complete?
--
Go to Headlight System Check

Check bulbs. had this happen on my intrigue had no high beams but lows worked. If bulbs are fine check fuses and relays.

Check bulbs and replace if nessasary. Always replace in pairs with quality bulbs.

This care is basically a Cadillac but without the badging.
Look under the hood for the main fuse block and then look for the headlamp fuses.
All four headlamps have their own fuse.

There are two fuses, one for right side head lamp and another for left side headlamp. Hi and lo beam on one side get voltage off the same fuse. The fuse circuits are hot all the time. On and off, hi and lo beam are controlled on the ground circuit. So, if you use a redundant ground tied into the low beam ground, then run it to a toggle switch located under or on the dash, if you want lo beams on, flip the switch. I'm not an electrical engineer, but that seems easier than using a relay.
If you want to talk relay, usually terminals 30 and 87 are the load side of relay. In a diagram it looks like a line with arrowhead on it, open, unless relay is energized. Terminals 85 and 86 are for the control side of relay, voltage and ground. If you have constant voltage and ground for the control side of relay, relay will be energized all the time, which you don't want.
Just my opinion. Whatever you do, you may have to check for unforeseen consequences?

If it's a 3.5l, high side is near the air cleaner, low side is at the firewall. Econo means compressor is OFF when light is ON (backwards). A low charge will cause hot air on left and cold air on right vents.

Turned out to be a 15 amp fuse not making full contact. Thankyou anyway gentlemen.

Cover the simple stuff first....you could have just blown both high beams. If the bulbs are ok then the most likely is a headlight switch.

Run's continuously ? High speed , low , med 1,2, 3 . ????
Probably the high speed relay contact's stuck closed . The fuse for the high blower is hot at all times , low blower fuse is hot only with key on. It sparks becaused the motor turns on . If it had a short the fuse would blow . The blower motor relay is part of the resistor pack near the blower motor .

sounds like the dip switch needs replacing or the high beam filiments are blown on your globes