When things get out of hand, you need to step away from the issue, and get a grasp of the situation. Sometimes when people come at your from different angle, all with urgent objectives, you need to plan out what can be done and act accordingly.
First create a list of everything needs to be done. It can be in on a piece of paper, on a post-it note, in a google document, where ever. This doesn't have to be perfect, its a starting point to weave through the chaos, and can always be appended. Once you have an initial list to work off of, time to prioritize.
Prioritize based on a few principles; timeliness, value of delivery, amount of work, risk, and consequences on delay. Timeliness is just when does it need to be done by. Value of delivery is how much value does it bring to your stakeholders on its delivery? Are people waiting on your part before they can start theirs? Amount of work is how much work it will take you do complete. Risk involves how much are you guess with all this? Is it something that could take 1 hour or up to 10 hours, this is high risk. Consequences on delay, if people are contracted to do something, but can't start without you completing your end, your consequences on delaying amount to $. Mapping these out the best way you can is ideal. It doesn't have to be perfect, and just thinking each of those out should clear up your priority.
If prioritizing didn't get your engine going by now, take the top priority item and break it down into its first steps to completion. Ensure you keep this list someone you won't forget about, because it may have to be revisited to expand. Once you have simple steps to get going, just blindly start completing those simple tasks. This momentum should get you out of a panicked state, and into a completion oriented mindset. If you complete those items and start entering a panicked state of what to do next, just plan out on simple steps. And yes, talking to someone about next steps can be a next step. You are never in it alone.