What you need to do now is trade your picks for similar picks In the year after next year. Reason is because any player you traded or released that has bonus money due to him, will affect your dead cap.
The key detail here to keep in mind is that any player whose contract has more than two years remaining with bonus money, will hit your current season’s dead cap for the single year amount shown in his contract. But he will then hit your next season’s dead cap for all of the rest combined.
For example... John Smith has 4 years left on his contract that includes a bonus of 1 million per season. If you trade him or cut him, the current season will have a dead cap hit of 1 million. But the next season will have a dead cap hit of 3 million. Therefore, players with more than 2 years left on their contract have the potential to choke your salary cap much worse in the next season than the current season.
Now that you have your draft picks, your still not out of the woods yet because they will still have big cap hits with their contracts. And as you currently sit, they will collide with your dead cap and put you way over the salary cap (as you already seem to realize).
Your dead cap will greatly diminish after next season. Therefore, you will have a lot of room available for the contracts that will come from draft picks and/or high priced free agents. That’s why I started off by saying you now need to trade your draft picks for picks that take place the year after next (the 3rd season).
Extra tidbit of advice.... For now, that’s as far out of picks you can get back in trade. But as the seasons progress, you may want to consider trading some of those for later seasons so you can spread them out just a little bit. I recently did all of this with a team and now am having a tight situation of extending everyone’s contracts. I didn’t have as many draft picks as you.