May not be your answer but here are a couple things that might be worth a try
1- Change the outlet that the power supply is plugged into, avoid using a surge protector.
2-Try a new power supply, if you have a friend with a Slim system as well that would be the best way to try it. If you are going to buy one just to test it I would recomend buying used so that if there is no change you can just take it back and get your money back.
3- Sounds simple but double check the connections between the power supply and the xbox itself clear any dust and make sure they are plugged in all the way. ( you would be surprised how many times this has been the source of one my electronic issues)
Hopefully one of those suggestions helps you with your issues man.