In short, yes the chain can need adjustment that fast, especially if it's starting to wear out, assuming you have a stock non-Oring chain. I recently changed my original stock chain on my 2005 TW. For the past couple years I had to adjust it nearly as often as you are mentioning, if not more. Seemed like after every ride. I should also mention my TW had about 6,000 miles when I changed it. I cleaned and lubricated it often, but towards the end of the chains life, it didn't really matter.
You can adjust the chain with the on-board tools if complete. Little harder of course using the wrench on the axle nut rather than a longer one from home, but can be done. I've adjusted mine out on the trail with the on-board tools several times. I used a rock to gently tap the adjusters to the next appropriate notch. Even though the chain is wearing out (most likely) you should notice a bit better shifting as well. BTW, I love my new o-ring chain. Much quieter and have not had to adjust it yet after several rides. Got to get use to not adjusting the chain as much as I use too!