It is very difficult to work under these "ready, fire, aim" conditions. It sounds to me like you are receiving requirements from a very insecure person, whose opinion changes each time a higher up person suggests a conceptual idea.
In these kinds of situations, I have found it valuable to wait AT LEAST an hour before responding to emails. (I'd ignore the texts, unless texting has widely replaced email by your organization as a whole.) Read them, maybe, but not respond. This way you can spend your time focusing on the actual work you need to do, not the discussion of random urgencies that may become irrelevant tomorrow, or even two or three emails later. In my last job, if something was REALLY urgent, someone would come over and talk to me in person, assuming I hadn't seen the emails yet (if you work remotely, a phone call with an actual two-way conversation may be the equivalent).
When you have the face to face or phone conversation, it is helpful to repeat what the person is asking for in your own words, and then ask your questions about the new requirements and priority. "If I understand correctly, you're saying we should stop working on Current Top priority X and now focus on Priority of the Minute Y. That's a big shift. Can you explain the change in the business? I might need to do more background work than just changing the UI. Will there be changes in other business processes, like billing or inventory (for example)? Are you going to expect these new data elements to appear on all the monthly reports?" It is also useful to say something to the effect of " You understand that if we proceed with this new effort, it will delay the release of Current Top priority X by at least a (week, month, fill in WAG time estimate here), right?"
If it's a real emergency, the requestor should be able to answer these kinds of questions, or immediately refer you to someone who could. If it's not a real emergency, this kind of conversation will force the requestor to slow down and determine how important the change really is, given that they need to get you more information. Often they'll see that what's already in the pipe is more important, or at least not worth stopping, and the new request can go on the list.
If the changes are determined to be necessary, I've found it useful to write down what was requested and your understanding of the changes in an email, and send it to the original requestor, asking if they agree on the scope of the change, again, as clarification. This way you have written documentation of what needs to be done, and why it was requested, in case there is any blow back on why you are no longer working on Current Top Priority X, or need to explain why the original deadlines are not going to be met.
This should hopefully improve your relationship with the requestor, since you are demonstrating your knowledge, and making sure that you are working on what they want, but you are being honest about what it takes to make changes. By asking about the request in detail, they see that you think ahead, and consider things they might not have originally.