Can I start my new job while on transition leave?

SGT Payton; the LAW (!) says you can... it is stated as such in 5 USC: §5534a. Dual employment and pay during terminal leave from uniformed services
A member of a uniformed service who has performed active service and who is on terminal leave pending separation from, or release from active duty in, that service under honorable conditions may accept a civilian office or position in the Government of the United States, its territories or possessions, or the government of the District of Columbia, and he is entitled to receive the pay of that office or position in addition to pay and allowances from the uniformed service for the unexpired portion of the terminal leave. Such a member also is entitled to accrue annual leave with pay in the manner specified in section 6303(a) of this title for a retired member of a uniformed service."
Anybody who tells you otherwise is clueless. You will notice that the law does not touch on civilian employers, to wit I state; how will DFAS rectify this? further more, if it is good for Federal employees, I don't think constitutionally you can prevent a person from working from let's say, his/her dad's family business.

SGT (Join to see) I wouldn't worry about it. Find a job, go to work. The only issue will be if you worked as a contracting officer and dealt out contracts to the job you're entering in on. Otherwise, get a job and go to work.

You can work in the private sector while on transition leave. There are a few limitations, but they mostly deal with working for a company doing business with the government. You can't award a contract to a company, then go to work for that company.