Is there a grace period after I put in an offer?

If you as the buyer have removed all contingencies then there is not a grace period. 

When you remove all your contingencies https://shopprop.zendesk.com/hc/en-us/articles/4408547355287-What-are-the-typical-buyer-contingencies-  in essence you are telling the seller you are buying the property as-is and it is not contingent on appraisal or financing. 

If you put in an offer that is accepted by the seller you are responsible for the earnest money. The question is whether the seller will come after you for the earnest money if you don't deposit or simply try to resell their home.