Engineer here.
There is an enormous amount of variety in engineering jobs, so take that into consideration. I would recommend engineering over teaching. The difference in pay is larger than you indicate because it's very typical for new engineers' salaries to rise quickly in the first 5-10 years. And based on what I see from my friends who are teachers the work life balance is much better with engineering.
Do you feel like your job provides meaning? The actual work I produce doesn't provide much meaning in my life. However, I was moved into technical leadership (not management) very early in my career and I believe I did provide a lot of valuable mentoring and guidance to the people who worked for me. I also got a lot of personal satisfaction from doing a great job by delivering a product that made my customers happy while also creating an environment where my team didn't need to work overtime or perform miracles at the end. It's very satisfying having a team that's happy with their work life balance and a customer who thinks you're delivering a great product. But the actual product itself? meh.
What are the best and worst parts of your job? Providing career guidance and mentoring to junior engineers and getting paid very well for an easy, low stress 40-hour work week, and solving technical problems are the best parts. The worst part has been the rare occasion when I've had a terrible boss. But I've been able to remove myself from those situations pretty quickly.
Work/life balance? Where I work, this is up to each person. Right now engineers are in very high demand, so it isn't a problem when they set boundaries. However, very few have whatever it takes to actually create a reasonable work/life balance. Before I FIREd, I know that any one of my team members who asked to work from home, transition to part time, get a raise, or pretty much anything else would have gotten it. I had one person who was a very low performer for their grade level, and when he decided to retire to move closer to be with his grandchildren we bent over backwards to try to get him to work remotely. It's better to have a low performer working remotely than an empty seat, which given the current environment is what we would have had. He was incredibly appreciative and stayed a few months longer, but eventually retired. Most people, for whatever reason, still seem to feel like they need to keep pushing hard at work and short-changing their real life. With the job market as it is right now, I suspect in most engineering workplaces if you're doing good work you should be able to ask for the world and get it.
What would your ideal career be? Being FIREd! Seriously, I had just about the perfect job for me - great management, near total independence, a great team, high pay, low stress, flexible hours, and I only worked an average of 35 hours a week as I burned down a high PTO balance for the past year+ as I eased into FIRE. But that's nowhere near as good as not working.
One last thing - at my company, there are a lot of opportunities to do other things that might scratch the "meaning" itch you have. My company supports the local science fair each year with organizing and judges, there are a variety of company sponsored volunteer groups, and your local library or school is probably looking for math tutors.