i think you will need to get a new visa in this case.Since you are graduating you will be no longer in student staus for the june and the july month. I believe you will not be applying for your OPT too after your graduation because you plan to leave the county after you graduate. So you will be in the period that US goverment asks its student to leave the country after his/her study completion. Getting admitted and I-20 for a new college can help you ,however not for exit and entry to US. I-20 can only guarantee that you remain lawful for your student staus in US.
Regarding, more than 5 months stay outside US. This is only for the summer semester. For the long semester ,if you plan to skip 1 semester, you need to reapply for the VISA.Reapply for the SEVIS and you need to come to a univesity as an initial atendee.That means you will be treated as an new student in US when you arrive here.You need to wait for 1 more year for CPT or Economic Hardship work authorization.
For your case, since your new I-20 will clearly mention initial atendance at the new university. You won;t be able to enter USA on your old visa.
These changes came in 2006.
Finally, explain what your problem is clearly to your international student advisor.He/ She will readily help you.
The above all information are correct as to my knowledge, If you find any error do drop a message for me.
Best of Luck,
Perrybhadra