If your camera has a scene mode that is called "high speed", "sports", or "action", you can try using that scene mode with flash.
Other than than, there is no way to adjust the shutter speed.
There are several factors working against you.
The T1 has a very weak flash (4.9 feet) and it will move the adjustments toward more light, which means slower shutter speeds.
All light weight, small cameras are more difficult to hold steady. If it has a viewfinder, using the viewfinder will help because you are holding the camera to your head; which is steadier than your arms held out in front of you.
Resting you hand on a solid object is a good solution, but not practical all the time.
..................
Future camera purchases:
Select a camera that has a viewfinder, a tripod fitting, scene modes that include "action" type modes.
Check to see if the flash coverage is over 10 feet.
The bigger, the better.
To really get control of the shutter speed, get a camera that has "shutter priority" and "aperture priority".
That lets you:
..select the shutter speed and the camera will select the correct aperture for you.
..select the aperture and the camera will select the correct shutter speed for you.
Cameras with these manual controls can be inexpensive.
i.e. Canon A510 ($199).
...
..
.
i have a sony t1 and when taking pictures inside with the flash on, the only way i can get a non-blurry picture is to rest my hand on something. this (i'm assuming) is due to the shutter speed of the lense being too low when the flash is on. if it is due to something else, please tell me. i also have an old canon digital camera and it happens very rarely with that one.
is there any way i can adjust some settings in my camera to eliminate or reduce that problem?
in the future, if i decide to get a camera with a faster shutter speed when the flash is on (without spending thousands on an SLR), what kind of camera should i look for (brand, type, specification, etc)?
thank you for answering my questions.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic