I have been there, but I did find time because the alternative was having too much time when the tutor program was cut or eliminated. You need allies and visibility both in good and bad times. In good times, those times when you are not fighting budget cuts, you need to let people on your campus know how well things are going and how many students you are helping. This way you will have developed a positive image of success and service, which will help to stem the tide of cuts and, maybe, even get increased funding. There is a huge source of support, experience, and successful ideas close to home.
Involvement leads to the second strategy, credibility.
If you have not gotten your tutor program certified, I strongly encourage you to do so, it is a very user friendly process that assumes that you will be successful in your attempt to be certified. If there is an area that the evaluators, all tutor trainers themselves who have gone through this process with their programs, believe needs improvement, the evaluators will provide useful suggestions and advice in order that you will be able to got your program certified. The certification process is intended to add credibility to excellent programs by granting them international certification and to help programs that are not get at that level to achieve that excellence. If your program is already certified, please remind people on your campus. You do not have to wait for your periodic recertification to do this, though it is a perfect opportunity.