I just tried that in a league...had 4 players I wanted to resign. I increased their asking bonus by roughly $50,000 and 0-4 signed. I know JDB was going to fix this so you couldn't sign anyone the last day. I may have missed a post in which he said he did this.
One thing I try and do is resign a player from his initial contract, usually in his 2nd or 3rd season. This usually will give you a savings, however, since the season is over...no more salary to be paid for the current year...this may be tough to impossible with no cap.
I have been in this situation before, tribewriter, and I did let the players go and signed a bunch (for the minimum) the following season. Since the team I took over was awful, I did get quite a few players off waivers the following season and it didn't take too long for them to be competitive. A lot of work, but I've seen some of your rosters in other leagues and I'm sure you will come out pretty good very soon.