Hello, I have been laid off from Amazon, and I am unemployed since March-20, I got one offer from another company, and they did my immigration assessment, they told me that I do not have enough time and its difficult to ensure that I will not go out of status within this time saying that they have rejected my application. If I just file LCA my unemployment days will stop, or they will have to file my H1B change petition only then will I be able to save my status. Also, will I get 60 days grace period for each H1B petition?
Filing an LCA is not enough. In order to stop the running of the 60-days grace period and to start working, your H-1B transfer petition must be received by the USCIS within the grace period.
The grace period is not a one-shot deal. You will get it whenever you are in the layoff/cessation of employment situation again.