Having an employee opens a whole can of worms that you can not cirumvent by going the 1099 route. Many do go that route and never have any problems but it is not the correct way.
The cost per employee is high but you may look at a payroll service to do everything for you. Many times your bank will offer this service through a company like paychex. This is the best way starting out. These services often include WC and handle all of the taxes including suta/ futa.
You do not have to have Workers Comp in Texas. However, you are subject to 100% of the liability and disability claims without limits.
You do need to report new hires to the state atty general too. Something about about dead beat dads.