Can I remove the relay for the ABS system and still run standard breaks ?

ABS warns you about a problem in your ABS System starting from the fuse, or relay and the other light is not for the hand break ! it's warning you about your break PADS.
So change your break pads and check your ABS system functionality.

check fuses and relays ---check your abs [anti-lock braking] system a sensor or wiring connection may have been disconnected or damaged while work was being done until fixed try removing the abs fuse this will stop abs control and give you standard brakes

The computer has detected a condition that indicates a fault with the ABS/Traction system. Typically that's caused by a wheel sensor problem, but it can also be caused by a faulty connector, peripheral device or the computer.
The above is the shop or garage answear. How about telling us how to fix the problem , besides replacing the computer or wheel sensor? How do we check these things?
Sometimes can be as simple as a short circuit in the rear break lights or a faulty connection in the break lights.
You can start checking the break light system. Then the sensors and finally the computer.

Antilock Brakes

Why not just disable the ABS? Pull the fuse or relay.

this could be a relay for the ABS system not turning off causing the box to hum.

the module is bad. needs a new abs pump. to temporary remedy it find the pump and disconnect its plug. be careful as the abs now is not working ( not that it was before) and incase of full breaking car will lock and slide tires. take it to a shop to get replaced.

I would, Double check your brake fluid level first. Also;

The water from the pressure washer may have made it into 1 or more of the electrical plugs on and/or around the master cylinder. You may try carefully removing the plugs, ( be careful not to break the little plastic clips that hold them in place,) try and use a hair blow dryer to make sure there's no moisture in either end of the plug and then reconnect.

As far as the ABS, ( anti-lock breaking system ), it is a safety and secondary part of your breaking system and usually would not meet emergency statis by itself, however, with your primary breaks also having a warning light on, I absolutely would consider your break issues as an emergency.

I hope this helps.

swap out abs relay with a similar known good relay it's possibly a sticking abs relay

For a short term fix unplug the ABS relay. Your ABS light will stay on, and you won't have Anti lock breaks but you will be able to start your car which is always nice.
If you actually want ABS they try replacing the ABS relay, this will probably do the trick

The ABS relay should be located in the fuse box located on the drivers side inner fender in the engine compartment. Lift the lid and check the location diagram on the inside cover to locate the correct relay/fuse.

A related issue may be a loose ground wire. There is one located on the top rear of the engine block. It is a bolted connection. Remove this bolt, clean the surfaces and replace. This simple step has cured many electircal issues on these trucks.