You don’t necessarily need paid experience. I’d argue that shadowing can be more informative about the actual job of a vet because as a paid staff member you have duties to perform, and those duties are not ones you will (or should, anyway) have to do as a veterinarian. Paid experience is great but having lots of shadowing hours is fine too.m
As for how to find experience, just keep applying to positions. For shadowing, I’d ask to shadow one day (low commitment for both parties) and if it goes well, ask if you can come back before you leave. Be willing to drive a bit. Talk to the vet you take your own pets to or that family members use. Talk to trusted advisers and friends and see if they know people. For my only paid clinical vet experience, my mom worked with the vet’s father in law and they helped make connections. For my main pre-vet experience in a diagnostic lab, my pre-vet advisor was married to a pathologist…she knew the lab needed someone and despite them usually only hired 2nd and 3rd year vet students, she convinced them to take a chance on me as a 19 year old pre-vet and hire me as a lab tech. That one experience fully changed the course of my life and is why I specialized in pathology.