Provided you never entered with immigrant intent, you may be eligible to apply to adjust status. But if your priority date is current, it may be faster to consular process as the adjustment process may take much longer, and if you had immigrant intent, it could be much riskier.
You will need to notify the National Visa Center that you are opting to file Form I-485, Application to Adjust Status. It is important to note that since you entered the US on a B-1/B-2 visa you probably should wait at least 90 days from your entry date to file for Adjustment of Status, to avoid being charged with misrepresentation. This is because when you last entered the US with your visa, you conveyed your nonimmigrant intention to stay in the US for a brief period of time as a tourist or to attend business meetings. Technically you are allowed to change your intent after 90 days according to the US Department of State. You should consult an attorney to verify that you are eligible to file for Adjustment of Status.
Yes, you can do an adjustment by filing the I-485 as well as other pertinent documents for employment and traveling. However, I suggest you consult with an immigration attorney to make sure your timing is OK to proceed since you are here on B1/B2.
As I recall, in a similar situation, we simply filed the Form I-485 with supporting documents including a copy of the I-797 Notice of Action verifying the approval of the I-526. As a courtesy, you can inform the Department of State that you will be Adjusting Status in the United States. Congratulations on the approval of your visa!
I would suggest you contact your attorney to file an adjustment of status application (AOS) for you, as long as your priority date has become current. Usually an AOS application should be filed 90 days after you enter the U.S. with the B-1/B-2 visa.
If you are in the U.S. in valid visa status, you can file for Adjustment of Status based on your approved I-526 petition. You do not have to pursue consular processing - if you pursue adjustment of status, you should notify the NVC to close out your file. I would recommend that you wait a minimum of 90 days before filing for adjustment of status given your entry in B-2 status. It is also important to note that once you file the application, you cannot travel abroad until you receive advance parole (travel permit) which is taking upwards of 8-10 months.