What happens if we go over our monthly submissions and decide during to upgrade our plan? Will we have access to all the submissions over AND under th

Yes, you will be able to keep receiving submissions once you upgrade your account, but note that subscriptions are automatically renewed and if you only want to upgrade to a month, you will need to request the cancellation prior to the renewal date, this way you're not charged for the next month. 

https://www.jotform.com/help/331-How-to-Upgrade-My-Account 

https://www.jotform.com/help/341-How-to-Cancel-a-Subscription-and-downgrade-back-to-free 

I hope this helps.