The grace period is for the stay in US only, but you should not have any break in your employment, so you can request your new employer to give your starting date immediately since you quit the previous job. The international office people knows this most of the times, otherwise you can explain them your situation and get the starting date so that you don't have any break in your employment. I know you have very little time, but this is important to stay in US. But how can one terminate the employment with 7 day notice? In universities one must give atleast 15 day prior notice. You get this information from your work place also and if needed you can approach the senior officials in this regard.
All the best.
