For my first paralegal role, I started the search just after I finished my exams, which I think gave me a slight advantage. Admittedly, I did not make any TC applications that early summer, because it was essential for me to find a job.
I was looking for a job by myself, mostly by setting up alerts for jobs posted on LinkedIn, TotalJobs, etc. For many paralegal positions all you need is a well-crafted resume and (sometimes, but not often) a short cover letter so whenever I found something interesting, I would apply straightaway.
Another thing that I found extremely helpful was making my CV to look distinct - I invested £4 in a nicely looking (but still professional!) CV template on etsy. That was when I started receiving responses to almost every application I made because, by admission of many recruiters, my resume was eye-catching.
In the end it took me three weeks to find a paralegal position and I was lucky enough to actually secure three different offers from which I could choose what suited me best. Anecdotally, my chosen role was with a company that required LPC in the job description, yet I got it anyway
It was different when I was looking for a new job nearly a year later - I decided to use agencies at that time, mostly because I was preoccupied with work and TC applications so it was easier to delegate this task to a number of agencies (a word of warning: while there are some recruiters who will definitely go above and beyond to help you and will stay in touch with you, sadly I also met some that could not care less on updating me on the process, or failed to send my resume somewhere, or - my absolutely favourite - forwarded my CV to one firm without my prior approval, allegedly by an unfortunate error, which in turn put me in a very uncomfortable situation).