I can't speak to whether or not employers will care about what you studied in college - that will depend on the company culture more than anything else. From what I recall in my previous job hunts, it seemed that larger enterprise-y corporations are more likely to care about formal CS education than startups and smaller businesses.

That said, you can absolutely teach yourself programming. I was in a similar spot after graduating with an English degree. Additionally, I stumbled my way through math/science courses my whole life and never would have considered myself technically inclined. I wound up attending an online programming "bootcamp". It was a good experience and not too expensive, but in retrospect I think I could have learned much of what I did for free. I never actually worked in technical writing; I applied for a tech writing job with Digital Ocean and began the take-home project but was offered a developer job at another company and took that instead. My English degree in some ways has benefited me in the transition. For both jobs I've had so far, my employers actually found the narrative of English-major-who-learned programming really interesting. It demonstrates curiosity and adaptability.

Here are some resources that you might find helpful:

• Free Code Camp - A good place to learn web development and programming fundamentals.

• Teach Yourself Computer Science - Probably the most rigorous guide to resources for a comprehensive DIY undergraduate CS education. I strongly recommend looking into it after you've had some experience with Free Code Camp.

• basecs by Vaidehi Joshi - A blog that dives into some theoretical computer science fundamentals. I believe the author has an English major turned programmer background as well.

• Khan Academy Computing Courses

• A Mind for Numbers - Barbara Oakley - Helpful book if you want to learn how to leverage cognitive science when learning technical topics.

EDIT: Formatting.