Do you have to open a new line to purchase an apple watch

No you don't.

If you purchase the GPS model, it will work without any cellular connection as it will use your iPhone.

If you purchase the Cel/GPS model it will work the same as the GPS alone if you don't have the watch added on your plan.

Most providers charge around an extra $10 so a month to add the watch to your phone plan.

No. The Apple Watch, if its the cellular model, uses your iPhone's line. You will have to add it to your iPhone's service plan and pay the extra monthly fee for it however.

The GPS-Only watch does not use cellular and there is no extra fee.