Is the line wound reasonably tight onto the spools? If it is then it sounds like your line could be twisted as a result of leads/rigs spinning during the retrieve and swivels not doing what they're supposed to do!
Take your rods & find a field (with longish grass!!) and lay the line out along the field in the grass. Simply reel in through the grass (dont walk towards the end of the line with the rod while reeling) & it should get rid of alot of the twists that are in the line.
Alternatively, you could try one of those Gardner "Spin Doctor" leads. I've never used one so couldn't comment on their effectiveness but I suppose they would work (That is if your line doesn't birds-nest when you cast it out!!)
I supose it wouldn't hurt to replace the line - depends on the overall condition of the line. If it has been tangling on the cast then there's a good chance that there are kinks where the tangles occured which might mean the odd weak spot.
Can't really comment on Braid as I have only recently started using it for Lure fishing, but suppose it would outlast mono.