2000 GMC Yukon Low range

My vehicle has a push-button selector for 4 wheel drive high and 4 wheel drive low. It will go in 4 wheel drive high fine but will no longer go in 4 wheel low. When I push the button it blinks as if it's trying to convert but nothing happens and it never converts. I was told by a local dealer that some sort of computer module had gone bad and that there were no real mechanical issues. However I was also told that the repair would be about $1200. With high mileage I'm not interested in spending that kind of money but I'm thinking I could possibly do the repair myself. Can you please tell me what part (name, item #, etc) is probably the culprit and what would be required on my part to replace it? Also, what would a good source be to find the part economically? Thanks and Merry Christmas!