If you're getting completely knocked by your resistance training then it sounds as if you may be working with too much weight. What sort of reps are you doing? I haven't done a huge amount of weight training, but when I do I work on 3 sets of 15 reps.
The other thing I find myself wondering is where your priorities really lie. Rugby doens't sound like a sport that combines terribly well with cycling! Every article I've read about weight training in combination with cycling says that having large muscle bulk is detrimental on the bike as it negatively affects oxygen take-up in the muscle fibres. Of course if all the sports on your list are there for your own personal satisfaction and fun, then you should be able to find a way to combine them all satisfactorily. But if your ambition is to be especially competitive in any of them, you may want to look at whether the other sports complement or hinder your chosen sport.
Perhaps you could considering changing position on the rugby field to something which goes better with your other forms of training?