What you are taking about is a rollover, of funds from one Roth IRA account to another. You want to do this by direct transfer -- open the new account and give them the account info of the old account and have the two plans transfer the money directly.
Rollovers and contributions are separate, and you can do unlimited rollovers without affecting your annual contribution limit.
However, if you withdraw the money and have it in your own hands, things get a bit more complicated. You have to deposit the money in the new plan within 60 days, and you have to tell the new plan it is a rollover. This type of indirect rollover is allowed once per calendar year. But if you fail to tell the new plan that the money is from a rollover, then the withdrawal will be taxable and the deposit to the new plan will count as a contribution, which will reduce or even exceed your annual limit. That's why it is best to do direct transfers when rolling over money from one plan to another.
