I have similar struggles as I take the same philosophy. I think you could also likely be getting hurt by the amount of early picks you have, based on the rookie reserve. Someone should correct me if I'm wrong, but having multiple 1st round picks pending means that when you go to resign your players that there is significantly less cap space available. I find myself in cap hell when I'm in your boat.
I tend to trade players that I know I will not be able to resign (had to once trade a 90 OVR ILB) for cap purposes. I try to flip those players for 1/2 round picks. I might be taking advantage of the CPU trade logic a little bit, but I enjoy nailing the draft. While this might be slightly unfair, I live with myself because Compensatory Picks do not exist in the game. I look at that as a way I even that playing field.
I also wait to develop the traits that most greatly affect the overall in some positions. For instance, RBs are major cap-eaters. I tend to focus only on their physical traits and just bank a lot of XP (thus keeping their OVR low) until I can resign them. Then I lock them into long term deals at a discount and splurge on their XP after that. This strategy works best if drafting 21 year-olds before they hit the dreaded age 26. Again, it's slightly hacking the game, but in real life resigning a player early would count the bonus towards the current year's cap space and to my knowledge that is not the case in CFM. These intricacies of cap management are just not in the game.
I don't know how much this helps, but I have had to spend time examining the players on my team. Be extremely wary of 1st round picks as they eat cap, as do QBs. I cannot seem to lock up a QB without eating up massive amounts of salary. I guess they have that accurate. I felt that last year was easier to resign, especially with the coach upgrade package. If you find a more authentic way to realistically re-sign players, please let me know! I understand big decisions, but there should be an easier way to re-sign current players on their rookie deals (as is usually the case in the NFL).