Just a note to back the Wilkinet. It's brilliant - the only baby carrier I could use without getting backache after the first 5 minutes. It looks a bit wierd at first, but you soon get used to it. I used that with dd2 all the time (she was a bit like how you describe - especially at certain times of day).
Another thought. I don't know if your baby is a bit old for this now, but have you tried cranial osteopathy? Again, my dd2 was determined that I would carry her around all the time, and it just wasn't always practical as I had no family nearby, another dd under 2 and when dd2 was 8 weeks old dh started a new job and was away on a training course for 5 weeks! We went for two sessions of osteo and she was like a different baby. Still a bit clingy, but it definitely helped with the constant crying. I was a bit sceptical, but I'm a convert now!
As for having a bath/shower. Just remember that it won't hurt your baby if you leave him to cry occasionally for a short time. This is what I had to do with dd2 during that period when dh was away - I'd put her in her cot when she seemed fairly chilled and then jump in and out of the bath/shower as fast as I could. I knew she'd be crying by the time I finished, but ... well... it was that or be stinky, so I lived with it! I'm not a fan of controlled crying, but I felt that in this instance it was a case of 'needs must'.
Good luck. Oh, and it probably is just 'a phase'. My clingy dd is now highly independent, and cries and clings much less than her older sister!