Yes, this is a perennial problem for me as well. I'm the type of manager that makes every roster spot count, which I think contributes a lot to my success in leagues even when I get a slow start. It's the reason I sit there chuckling when people are boasting about being 6-0, when really they're one injury away from a lost season. I get sick when I see some otherwise very competitive teams carrying 2 kickers or 2 defenses - but then happy because I realize I have a deeper bench because of it.
But obviously, this makes it difficult to cut players if you don't have any wasted space.
I tend to drop my kicker every week for the WW pickup early (i.e., this week it was Anthony Thomas), just to see how serious the injury is, how it plays out, etc. I'll try to package 2-for-1 but that rarely works.
At least it gives me the full week to figure out value, instead of making the decision on Mon. (And at the very least keeps the other guy off other rosters in the meantime if they're a short-term injury play).
Another thing is to try to know how the league values certain players. I know in my primary league, which is now 12 teams after a few years at 14, RBs are incredibly overvalued. We play 2 plus a flex, and people have been "trained" after the 14 team years. If I drop a RB, he will be scooped up immediately.
You also have to consider the other rosters. If some of the more dangerous teams are good except for, say, TE, you can carry two mid-range TEs because there's more value in keeping a potential starter away from them. This is one reason I have Watson rostered behind Gates and dropped Williamson and Michael Clayton in that money league even though WR is my weakest spot. My stance is that both are probably of marginal value to me, but Watson is more valuable to keep away from the competition.
And lastly, don't get too attached. Most sharks will have a great draft and then sometimes miss out on a guy like Boldin in his rookie season or Colston or Berrian because they think it's a one-time flash in the pan. The trick is gambling just enough with your depth and bench, balancing the upside with the solid base production.
(Sadly, I actually HAD Colston in my main league and dropped him for Watson when Watson hit the wire, but that's a separate story

).