The short answer is: yes, but it could create complications. There's a page from another University (not mine) that outlines this process: https://iss.washington.edu/procedures/change-status/b-1-b-2-to-f-1/
I would highly recommend you consult your new school's international student office and follow their suggestions. They would know best if people from your country will typically have trouble making the switch from B1/B2 to F1 while in the US and/or if traveling to a 3rd country (e.g. Canada or Mexico) and re-entering would be straightforward or not.
After consulting your school's office, you will have to determine which choices and risks are acceptable to you:
1. Rebook your tickets to the US to arrive on August 1 or later. Note that for many airlines, even "non-refundable" tickets are still cancellable and you can still use the funds paid for that ticket for another ticket with the same airline, minus a cancellation fee (usually $200-$300). It sucks to lose this money, but it might still be less than the costs of the other two options. So I'd consider that. This would be the most straight-forward and surefire path.
2. Enter the US on July 12 as planned, then go to Canada or another country after August 1, and re-enter the US on F-1 status. Off the top of my head, the additional costs are the cost for the trip to Canada and the potential risk that the border agent will wonder why you are entering from Canada instead of your home country. However, it is fairly common for international students from faraway countries to apply for their second (or third or fourth...) F-1 visa from a US consulate in Canada or Mexico and then re-enter there instead of going all the way to their home country to get a new visa. My school's international office presents this as an option but also points out a risk: if your visa application is delayed, then you cannot re-enter the US and you are "stuck" in this other country. However, since you would already have the F-1 visa, then this risk may not apply to you.
** There's also option 2b, where you go to the US on July 12 as planned, then return home to your home country and then fly back to the US again within 30 days. This might make sense if there is some reason why you really wanted to be in the US on July 12
3. Finally, you can also just stay in the US and apply for a change to F-1 from B-1/B-2. The main cost is the filing fee, of 370 USD, in order to make the change, according to https://www.uscis.gov/i-539. In addition, US Immigration isn't known to be very fast on their paperwork. So, even if you apply for the change the day that you are eligible, there is a risk that you won't get it approved before your program starts. If you are not approved, then you cannot start your program! So this could be very bad!!
Again, please talk to your school's international office and find out more information, in case my info here is wrong or outdated. But, given the high cost of option 3 and the risk of delay, I would personally not want to do that at all. Option 1 (just rebooking) makes the most sense unless you are going to lose a lot of money or you have another pressing reason to be in the US by July 12.
