Does anything have to be filed with the state or is it just a matter of following the operating agreement?
Your answer will most likely be found in your operating agreement (aka company agreement). The operatiing agreement should address the process for removing a manager, if the LLC is manager-managed and should also address how you would add new members. Typically, the managers are voted in and can be voted out. To add a member, you would typically just transfer/assign a interest (membership interest) to the new member. The operating agreement may require approval from the other members or the mangers.
If your operating agreement is silent on these topics (or you do not have an operating agreement), the Texas Business Organization Code will control.
Once the formalities are addressed, you may need to amend the certificate of formation on file with the Secretary of State.