What to do when you plan to submit a paper and a senior colleague is independently pursuing the same research idea? I am an assistant professor in a business discipline. A couple of months ago, I came across (through casual chat) that a senior colleague of mine plans to pursue a research idea that I myself was also planning to pursue. I asked whether I could join her and her co-author on the project and she declined. But she said I could work on it myself. We didn't share any details of our thoughts afterwards. So we don't know what exactly each other's research question and progress on the project.

I went ahead and explored the project with the research question I had in mind. Now I have a finished draft of my work. I don't know the status of my colleague's project. 

Now I would like to submit my paper to a journal. But I am not sure whether I should publicize it by e.g. posting it online/on my website. I am afraid that my senior colleague (she will be on my tenure committee) will be pissed off if I do so. She may think that I am scooping her. On the other hand, without publicizing it, people in my field (or potential referees of my paper) will not know that this work is mine and the paper may not get exposure/comments (which may not be critical). 

What should I do? I don't think I have done anything unethical, but somehow I feel that I am stuck in a dilemma.
The fact that the colleague in question will be on your tenure committee seems to, on the surface, complicate matters. However, it's not like you had a conversation with her, made no mention of a possible collaboration, and took the idea and ran with it: 

You reached out to her by proposing a collaboration on the research topic, which she declined.

Thus you should do with your manuscript whatever is normal in your field to disseminate the ideas contained in your paper; e.g., submit your paper for peer review, post the manuscript on your webpage/upload the manuscript to a preprint server (if this is normal in your field/OK with where you plan on submitting your manuscript), etc. 

Also, if your conversation with your colleague was key in your research progress on the topic, then I think an acknowledgment in your manuscript stating that fact would be appropriate.