No question: adding an external flash. See previous question Prime lens or flash: which upgrade will most improve baby photos?, which covers some of this. A flash can freeze motion, and makes it easy to get enough depth of field to get the whole scene in focus. And when you can move the flash off camera, you can create nice light where it doesn't exist already.
You note that you feel like an external flash doesn't feel practical for most shots. It's actually pretty easy in most situations — put it on a small table stand, point it at the ceiling, and put it somewhere convenient in the room. That means you need something with wireless control, but luckily that's pretty easy to do these days — and much cheaper than going to a full-frame body.
For Canon, unfortunately the T1i's built in flash can't act as a wireless controller, although newer Rebel models like the T3i can. There are couple of recent low-cost flashes like the Cheetah Light V850 or the (upcoming) Cactus RF60 with integrated radio control. Unlike the optically-triggered remote options, you have to set the flash power manually, which actually isn't very hard when since lighting inside doesn't change very much, and these flashes actually let you set the power from the controller on the camera body. (And due-to-be-released-really-soon-now controllers will actually allow TTL with Canon or Nikon bodies.)
Additionally, if your kids are old enough to sit still for you for a few minutes, getting a few light modifiers will allow you to do neat things you couldn't do otherwise. I don't mean a cheap gimmicky thing, but an umbrella or softbox that's at least a 20". That doesn't help with chasing kids, but if they will sit for a bit you can do stuff like these:
These are straight out of camera; no time for post-processing, although next time I will comb her hair a little bit for the second shot. They're at f/5.6 — nothing you can't do with the lens you already have.
This is with the Cheetah flash and a Westcott 10×24" softbox really close to the frame. You can also use a $15 shoot-through umbrella, but the softbox is nice because it takes up a lot less space. That plus a stand and the flash is less than a nice lens would have cost — and for less than the cost of a splurge on a body upgrade, you could get two flashes, plus stands and with a large softener for each.
The catch is that it will take a little bit more work to figure out what you're doing — the often-repeated advice to go through Strobist Lighting 101 and Lighting 102 is really worth it. But if that sounds like too much work or stress, a single external flash and remote trigger of some sort will get you great results.
Or you can even leave it on the camera and bounce off the ceiling — even that will get you better results than pretty much anything else you can do for the money. See this photo from the question linked above (same subject, a few years earlier):
That one was with a flash pointed at the ceiling, with the built-in bounce card pulled out for the sparkle in the eyes. Notably, this wasn't done with an DSLR: it was taken with an Olympus compact; it's the flash that matters.
(All that said, adding the 50mm f/1.8 won't hurt either.)