In my university, which is in Canada, TAs are part-time positions for graduate students as far as I know. So do full-time TAs exist like other faculty members?
I'm aware of at least one R1 University in the US that allows some graduate students to be full time (100% equivalent appointment as a TA) during the summer, if they have no classes or other position, and if teaching needs require it. But in any case this was considered quite rare. My understanding was that the typical case was when the student was to be an instructor of record for a course, particularly one they designed or won a grant to develop, and usually they were also part of a teaching-preparation/certificate program. Even then the person would generally only hold such an appointment for one summer only.
At least in the US, part of the reason why this isn't more common is employment rules - a full-time teaching assistant would be a de facto employee of the University, and entitled to full rights and compensation for that time period, unlike the odd "not exactly just a student, not exactly an employee" category that graduate students possess in the US.
As GEdgar noted in comments, there are full-time people who are not full faculty in the US, but they are not called a TA (which is a title reserved for graduate students doing work part-time while also being a student); these people hold titles such as Instructor, Adjunct, etc.
According to my knowledge, No. As part of your graduate student funding package in Canada, the TA Fellowship is a part-time position for the semester and/or semesters in which you are appointed. Note: You can TA several courses. Note: In known cases, you may be given the position of the lecturer (if you adequately prove your capabilities) during your post graduate studies. I believe the pay is twice that of the TA position.