I've been a member of the ECF in the past. I'm not now because I have issues with the way the organisation is run.
There are four levels of membership:
To play in the sort of tournaments you want to play in you'd need silver membership. The only difference between silver and gold is that the latter enables you to play in FIDE rated tournaments, which are actually quite rare in Britain because the required time rates don't suit the weekend tournament structure very well.
Quite a lot of tournaments are listed on the ECF website:
But there are some which aren't.
Depending on where you live, you might find that evening league chess is an option. That's the only over-the-board chess that I play, I've only played in two tournaments (both rapidplay) in the last 25 years!
Edit: I forgot to mention that long play tournaments in Britain almost always have at least three sections, often named minor, major and open. There are generally five or six rounds and run on a Swiss system, meaning that you play in every round and will be paired against someone with the same number of points as yourself (or as close as possible). I've gone through some of your games and I think you'd be perfectly OK playing in a minor section and would probably win at least a couple of games.
Rapidplays, where each player has half an hour on the clock, are usually just run as one section, so you might find yourself paired against a very strong player in the first round. The whole tournament (usually six rounds) is played on one day.