I need to collect some personal contact information from my participants, as well as run an anonymised survey. Is there a way to accomplish both within LimeSurvey in one streamlined process?

My idea was to create two separate surveys for this and link them together, so participants are automatically directed to the beginning of the anonymous part from the end of the non-anonymous part. Is this a good strategy to resolve the issue? What would be the best way to go about that?

As you need to separate this, I assume you don't want to connect the two databases later, correct? So I think your approach is correct. Include the link of the second survey as end url and you can redirect respondents directly to the next survey. They might not even notice that it is a separate survey.