I keep trying to manually write code that I'm proud of, but I can't. Everything always needs to be shipped fast and I need to move on to the next thing. I can't even catch my breath. The only thing allowing me to keep up with the team is Cursor, because they all use it as well. The last guy that refused to use AI was just excluded from the team.
This is the problem. It's not new that a company rushes its devs to deliver new features at a pace that results in garbage code. What's new is that devs who are willing to can deliver those features fast using a LLM. This obviously looks great to the imbecilic C-suites. Deliver features fast, get to market quickly, and spend less on devs!
This is just short-term thinking, and it looks like you've noticed this. The team you're on won't change because the culture at your company is to deliver the next feature ASAP and focus on the short term. This is common with startups, for example, because it's a constant race to get more funding. However, it always results in some half-assed product that inevitably needs to be rewritten at some point. With LLMs now, you'll also have a team of people who don't even understand their own code, making it take even longer to fix things or rewrite it later.
Anyway, if you hate it, start applying places now. At least in the US (where I am), the job market is ass. The more time you give yourself to search, the better the chance is that you'll find an option you like.
This has always been an issue. From my experience, the best way to get in was through internships, co-ops, and other kinds of programs. Those tend to have lower requirements and count as experience.
Of course, today, things are a lot different. It's a lot more competitive, and people don't care anymore about actual software dev skills, just who can churn out SLOC the fastest.