I actually became both. I'm primarily a graphic designer and digital illustrator, but I also do web design. I'm very happy with my choice. I love my job and the work I do. It never gets boring or stale and every month I get to work on something completely different.
There are things I don't like about it some days but every career has those. No career is perfect, but the saying "do what you love and you'll never work a day in your life" is spot on. I couldn't imagine being happy doing anything else.
And you can definitely have a good stable job as either. I have a good salary and the company I work for has above average health benefits. I'll warn you that there are a lot of disgruntled designers who will say they can't find a job or it's a dead end or they wished they had gone to school for something else. You have to take what they say with a grain of salt. Some people aren't very good at marketing themselves to employers, some are just unlucky, and unfortunately some are just bad designers and don't know it.
The only thing I would say from personal experience is that the design jobs are shifting. There are a lot more web design jobs out there than graphic design jobs. The web is getting more sophisticated and more companies are needing web design services, not just for websites, but for web applications and web based mobile applications. Print is slowly dying and designers who position themselves for web based design are going to be in the most demand in the coming years.
Hope that helps :)
Graphic/Web Designer / Digital Illustrator - 7 years