I wouldn't be too worried about your competitors seeing that you're hiring. Can be a positive thing (things are going so well we need more staff) and even if not, people leave jobs all the time so it's no big deal.
Not sure you can just advertise a job on LinkedIn though. Their recruiter tool is geared around a yearly subscription with a flat fee for use rather than a job board style price per listing structure. You need volume to make it work, but if you have the volume I know it saves money (because I've seen their data and it's really good).
Do onto spend £1200 on a local paper. Can't see that being remotely viable these days.
Job boards have their place and will be very cost effective. Try the usual suspects like Monster, Totaljobs etc and the job boards associated with the broadsheets as they are also viable options.
Word of mouth,networking and informal advertising on social media though (including LinkedIn) are probably the best way to go initially.
If you're constrained by time though, and need someone quickly, then a recruitment consultancy will definitely help to shortcut the process. I can understand your reticence but you might be able to manage the experience by managing the brief and the contract better. It should only be expensive if you pay the fee having hired someone and then they leave but usually the fee is only payable after they have been with you a given amount of time, say three months. That should give you time to evaluate them and pass them on their probation.