Upvote:3
First of all, a clarification on the terminology. US "visas" are solely for entry and the expiration date of the visa just means the last date you can use it to enter. It does not have any relevance to how long you can stay in the US. When you entered, you were granted a J1 (probably electronic) granting you J1 status and an admit-until date. For J1, most likely you were admitted until "D/S", not a specific date, which means there is no definite end date to your J1 status. You remain in J1 status as long as you have a valid DS-2019 and remain in good standing with your J1 program, and you also remain in status for 30 days after completing your J1 program.
I will assume you mean that your DS-2019 expires today, and so you remain in J1 status for 30 more days. If your J1 does not have the INA 212(e) two-year home residency requirement, you could file I-539 for Change of Status to B2 visitor status. If approved it can grant you 6 months of B2 status. You can stay while the I-539 is pending, even if your J1 status ends. On the other hand, if your J1 does have the INA 212(e) two-year home residency requirement, then unfortunately, (unless you get a waiver of the requirement) you are not allowed to do Change of Status, and you have no choice but to leave the US.
Whether you have a B1/B2 visa is irrelevant to your stay inside the US, or to your ability to change your status, because, again, US visas are solely for entry. Having a B1/B2 visa helps you only if you leave the US and wish to re-enter in B1 or B2 status.