Unfortunately you do have to join the ASA through a swimming club to enter. Most clubs are happy for you to join them even if they do not run masters sessions. It is a bit of a nuisance having to go through them although once you are a member you can enter most competitions yourself. I wanted to do the open water championships this year but I didn't because that involved entering through the club although for a fee non - ASA members could enter independently.
The ASA really needs to come up with a better system for masters swimmers.
As for stroke length, to go faster you may need to shorten your stroke. The key is to find the right stroke length/stroke speed combo for you. Swimming fast does require a lot of energy in that you are moving fast (just like running opposed to walking) but should not require a lot of extra strength. If you find you need more strength your stroke length is probably too long. To get fast you need to be relaxed.