re: the swimming
Try daily lessons for two weeks instead of the 2x/week deal. I have found that my kids make almost no progress with weekly or bi-weekly lessons.

My kids have average coordination. I was like your son. In elementary school I was ALWAYS picked last and the team that got stuck with me would groan. I also have trouble remembering which light switch works what light, operating machines, parking in a tight spot, still can't throw a ball....BUT I am an above-average athlete now (IOW, I'm a better athlete than the average middle-aged mom). I am still slow when it comes to acquiring physical skills, but the funny thing is that I CAN acquire them, it just takes me longer. By pursuing a sport that emphasizes endurance I was able to be a good athlete by HS (recruited by colleges) and compete through college. I am still slow but I have a lot of stamina. Physical activity brings me a tremendous amount of pleasure and I am glad that I was able to (mostly) overcome my innate lack of coordination. If the swimming is palatable to you, I would recommend pursuing that. But I think any sport whre they can learn at their own pace (i.e. not a team sport with a ball) would be a decent choice. Tennis, martial arts, running, rowing...