Hard bit of food jammed into fingernail bed. finger swollen now. okay to wait a day or two before seeing Dr?

It's very painful. Started this morning when I was washing a bowl to make breakfast. There was something hard sticking to the bowl so I went to scrape it off with my nail, and I'm a nail biter, so my nails are very short. Needless to say, the hard bit of food from the bowl is now lodged into my nail bed. It hurts very badly (scale of 1-10), it's a 10 hands down. Tried to remove it, but it's very small and almost the same color as my skin, so hard to see too. Couldn't remove it. At work now and finger is swollen and more painful than earlier. Typing is hard to do b/c of it. Don't have sick time available at work right now, so can't leave. Even though finger's swollen, is it okay to ice it every hour or two to keep swelling down and wait it out a day or two to see if it gets better? Or since it's already getting worse, should I go to the Dr? I'm guessing they'll just give me an antibiotic for infection and send me on my way anyway. Any advice?