Nancy, I am not that adept at LinkedIn, but I don't understand why the non-paid version would not still give you the opportunity to connect with people? I may be an exception, but I love it when people reach out to me on LinkedIn to connect, and I have mentored people who reached out and simply said they were looking for some guidance. I think the key is in the approach. If you say "will you be my mentor?" recipients may find that a loaded question which implies a long-term commitment. However, if you position it as "I read your profile, and I was really impressed with your career trajectory. I am trying to advance in (named field), and I wondered if you would be open to a brief phone call so that I can learn more about your career." Of course, personalize it to what you're comfortable with. There may be plenty of people who won't respond, but all it takes is one or two to get the ball rolling.