Lots of Options
You might want to switch to campsite 24, if you are looking to do a backpacking/fishing trip. Its up the Little River Trail from the Elkmont campground. You need to make a reservation, but my guess is that not many folks will be up that way. The weather is quite fickle right now, but hopefully by the end of March it will have improved.
For reservations of backcountry campsites from the nps.gov site:
If your itinerary includes a reserved site or any shelter, you must call the Backcountry Reservation Office at (865) 436-1231 to make reservations. The Backcountry Reservation Office is open from 8:00 a.m. until 6:00 p.m. daily. You may make reservations up to one month in advance of the first day of your trip.
If you had your heart set on fishing Abrams Creek, you could stay at the Abrams Creek Campground (from where you would have to park your car, anyway) and hike up to CS#17 to fish. The Abrams Creek Campground is actually fairly small and not too bad (as far as front-country campsites go). You could also camp at CS#1, which is a small site, but not too bad either.
Another option is #18 on the WP of LR. The fishing is not as good, but its quick to get to (1.8miles hiking) and the site is nice.
My money would be on #24. Its a bit of a longer hike, but the fishing should be good and the trail is relatively flat, so you should be able to get there in no time.