Welcome from PA! I was in a similar situation and found a 1200 (would've take either 883 or 1200) and can say that I'm very happy. Enjoy whatever purchase you decide to make and $5000 should get you plenty of bike for the $.
I paid 3500 for my 03 sportster xl custom 100yr aniversary edition.
All it needs is tires and the carb rejeted because the guys just added
V&h pipes and screamin eagel AC. So i think i made out ok. You will
Welcome aboard. Like stated already, I would get a 1200. Shouldn't be a problem in your price range. Don't worry, it wouldn't be an bigger, heavier, or harder to ride than the 883, just more power. An 04 would be nice with the rubber motor mounts, but an 04 1200 might be harder to find for 5k. Oh, and my shop had a couple sportsters awhile back that have 135K+ miles on them. They were in fantastic shape.
Welcome from Nashville TN. You will love your Sportster and like the others have said, 5K is plenty for a nice, low mileage bike. Don't give your money away, shop around, E-Bay and Craigslist before just going to the dealer and dropping your dough.