Because you are in status right now, your employer or employer's attorney would just complete the H1B as a Change of Status and Extend Stay request. Do not leave the U.S. If you leave, it will need to be consular processed and you will be subject to the 100,000 fee.
You do not do anything at this point, this is all on your employer and their attorney. Since your employer took part in the lottery they should probably be able to complete the i129 themselves, it's very straightforward once the position merits Specialty Occupation. source: I file the i129 H-1B forms my employees at my organization, only difference is we are not subject to the CAP Lottery.