On my 2005 chev avalanche 1500 the brake lights stay on the whole time that I drive.

A vacuum leak would cause the check engine light to come on and throw a code. It is more likely mechanical. Ideas in the link.

https://www.google.com/search?q=2005+Chev+Avalanche+Z71+-+whistle

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loose connection some where

Do you have a ABS Light at the same time?

2005 Chevrolet Avalanche 1500 Service & Repair Manual

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There should be a brake light switch under the drivers side mounted on the brake pedal it may have broke or come out of adjustment in any case you could push the switch in by hand and tape it as to not drain your battery until you can get a replacement part for it if indeed it needs to be replaced otherwise it may just need to be adjusted.

it may be the brake light switch. have you tryed making the connection manually? or is it the 'e' brake switch staying on? try pulling up on the emergency brake pedal to insure the "e" brake switch is'nt bad before you replace a suspected bad brake light switch on the steering column. if you suspect a bad brake light switch,pull up on the brake pedal and look to insure the brakelight swich disenguages itself,it may only need adjusted.the plunger may be loose. or the plugin may be loose. if not any of these,then i suggest you replace the brake light switch on the steering column,at bottom,near floorboard,on the steering column itself.

Hi, this is not a terrible job, but you are correct, the axles need to be removed, After this, you remove the park brake shoe, cable, and hardware.Unbolt the shield, swap over park brake stuff, and reassemble. Not extremely har, but is a b it time consuming. Good luck...Coop

Any codes? Try first to check could be a code.

See if the turn signals work,and the flashers,the problem,is most likely in one of the flashers,they are stuck,the flasher.

Chevy trucks are known for wiring trouble most likely there is a wire shorted out under the carpet. Always test first though before ripping out interior lol