I was going to make a couple of suggestion, Brady, but now in the knowledge that you're in N.Ireland they no longer apply.
I can't imagine for a moment that the Tesco honey is anywhere near raw but if it's working for you - great.
Obviously your setups there may be totally different to those that we have but it's worth trying to find any small time local bee keepers in your area. You may be surprised, as I was, that there are actually quite a few even if just hobbyists. These can be found by keeping a watchful eye out for signs people put outside their houses, searching newspaper/shop classified ads etc. I found the best source was to go to my local county beekeeper association. They were able to supply me with a list of ALL the beekeepers in my area as it's a requirement to be registered I believe. I was then able to contact them individually and narrow down the potential suppliers I'd consider (non sugar-syrup fed bees, organically raised, unheated etc).
Also, as Tyler has mentioned many times, if you can find a source of honeycomb the likelihood that it's genuinely unheated is far greater than jarred honey. I live in a very rural region despite being in a small City and, personally, I'm fortunate to have found 3 different boxed honeycombs from local health food shops. This is worth investigating also.
I hope that helps!