Is there a trail nearby that crosses the river without having to drive in a car?

Yes, during peak season there is a foot bridge that is put up... off season your on your own (ask locals)

Yes, just south of here after you enter Henry Cowell park, most locals use the train tressel to cross the river. 😊

I was just there March 15 16 and 17th at the santa Cruz redwood rv park and we had no way to take our dogs across safely.