You are mostly correct. There are two sorts of DAO tokens now, on the ETH blockchain, and on the ETC blockchain. Your names of DAOH and DAOC are fine for now, so I'll use those terms.
Your DAOH are all fine on the ETH chain; that was what the hardfork was for. They are in a place where you can take two consecutive actions and have them back in minutes, cashed out as ether.
Your DAOC on the ETC chain are in an entirely more complicated, and compromised state. Since The DAO was exploited on 20 June 2016, the DAO hacker has control of the exploited 3+ million Ether and thus you don't have what you think you have.
There are other complications: the nearly 8 million remaining ETCeth that the hacker did not have access to, were put aside for safekeeping by a "white hat" hacker group that called themselves the Robin Hood Group. They did that to secure DAO token holder's (DTH) ether on the main (ETH) chain. It is not yet clear exactly what they will do on the ETC chain. However, reports are in today (in the past few hours) that someone with access to that account, ostensibly the RHG persons, have removed the approx. 8 million ETCeth worth of DAOC in that segregated childDAO and put the some 7.9 million ETCeth in an account.
If this all checks out, then perhaps you and other DTH will get approx. 2/3 of your DAO value back out on the ETC chain as well.
Good luck to you.