I’m thinking that because it happens when you’re flossing, that you’re just getting rid of crap that’s been living in between your teeth for a while. It’s happened to me too, when I’ve started flossing after not having done it for a while. It should lessen with time. Floss before you brush, and use a mouthwash like Listerine as well.
Those above have also given some great advice – like @mcbolden said, Waterpiks are super awesome if you can afford one. They shouldn’t be intended to completely replace flossing, though – if you get one, use in addition to flossing.
Cleaning your tongue is also a great idea, as @SavoirFaire said. I’m not sure if you can buy a tongue scraper commercially – I got one from my dad (a dentist) that’s made by Oral-B. My dad is very much against the use of the scrapers on the back of the toothbrush heads – apparently they don’t do nearly as good a job as the separate scrapers (just one dentist’s opinion). But if you can’t find a dedicated scraper, using the back of your toothbrush or just brushing your tongue is indeed better than nothing.
@lillycoyote mentioned an electric toothbrush – those are also super good at cleaning around the gumline. You can get them nowadays for no more than $10 – I’d suggest, though, getting one made by a dental company such as Oral-B or Colgate rather than a cheaper one made by who-knows-who.