To be honest, you just look a bit knackered to me. It doesn't strike me as a shockingly bad gait, and if you'd asked me to pick out someone from the video who was bobbing from side to side, I wouldn't have picked you out.
So yes.. working on the core can help. Also, you could work on running drills, easily found on youtube... and so help you maintain good form.
Hi Joy... obviously running clubs must differ.. but the vast majority are friendly and of very mixed abilities. Maybe you'll be the youngest. But so what - and anyway, there will probably be a sprinkling of under 21s there.
It's easy for me to say just go for it... having spoken to many people who are very confident club members, but who admit that they were terrified at first. Of course, there is no real reason to be terrified, but it's part of human nature.
I'm sure your local club will have a facebook page, which could be a really good way to get in touch, make a few enquiries about what they're like and just 'chat'. Or at the very least they'll have an email address on their website.
In our club, if someone enquires before turning ujp, then we're expecting the newcomer and make sure we look after them on the first visits until they're bedded in. We also allow runners to run a few time, to make sure they feel comfortable before they sign up as a full member.
It sounds like you really do want join up... so find a way to make it happen! Good luck.