Yes, you need a sense of design but some people are making it out to seem like you have to be an artist! No! You can be skilled in no arts at all and be a great Flash designer for example. I'm in a Multimedia Design program (3 year) and web design is just a part of it...we do everything from web to DVD production (interactive menu's, DVD authoring, etc.) to video editing, sound design, etc etc etc. and me and some classmates often go to these parties where a bunch of web and flash/actionscript guru's just get together, have some drinks, and talk about their work (and even work on it). These guys are amazing and you can learn a lot from them, but they always emphasize that being "artsy" is NOT a requirement. There are no design standards...almost any creative level or design style can work on the web, so don't let people turn you off if you're not an "artist."
Also don't get discouraged about the lack of jobs now...everything grew too fast and suddenly everyone and their grandma had their own little design companies. But now little Joey down the street who runs the web design company out of his bedroom is out of work and the true Pro's are taking over. In 3-5 years the market should be VERY nice if you have some good skills (it will be competitive though).
good luck
p.s. you are really opening your options up taking both graphics and web programming, but is there one of the fields that you'd like to specialize in (design vs. backend programming)? You could save yourself some time and money if you were a little more clear on your goals here.
p.p.s. as for the work itself, I really liked it. I worked in a web design firm for about 10 months full-time before coming to college. you don't have to dress up to go to work (unless you consider some decent khaki's and a nice shirt "dressing up"), i worked my own hours basically. some days i'd be in there for 3 hours, some days 10. as long as i had my 40 hours at the end of the week, they didn't care. relaxed atmosphere, got along with everyone...pretty good experience. i don't know if i'd go into this as a career though cause i'm after other things. my goal is to work at some major movie studio producing DVDs but if that fails i got some other stuff to fall back on.