Btw: An alternative to using changing IVs is to use a fixed IV, but encrypting and including 16 bytes of strong random at the beginning of each datagram.
This will make sure that a re-used payload, key, and IV, still leads to a strongly different ciphertext. This allows you to use CFB mode for each packet, without any inter-packet depdendencies.
Probably yes, but: (a) you will be transferring these additional 16 bytes per packet, and (b) I am not sure that crypto-guys will like it (technically, without inter-packet dependency, it won't be a CFB anymore, but a highly-insecure ECB; while you do address the key problem of ECB, on the other hand all the modes are very fragile, so any deviation is generally frowned upon in the crypto-community :-( ). In other words: I do not see big problems with your approach, but proving it would be a very different story :-(.
Btw: When using CTR mode for UDP datagrams, don't you have to bump the serial number by the number of 16-byte blocks for each packet you send?
Yep. In practice, you can reserve like 12 lower bits for block size (for UDP, going above 2^12*2^4=65536 is not possible anyway), leaving you with 2^116~=10^35 packets (and at a rate of 1000 packets/second it will last 10^24 or so years - well, I can sleep reasonably well with such a limitation ;-)).
If I use a DTLS lib I don't have to worry about encryption at all?
Yes, and you can forget about sequence numbers too (DTLS will do it for you). What you DO need to remember about, however - is to make sure that you DO run your own CA (certificate authority - it is as easy as running a bunch of scripts on a dedicated laptop), you DO issue a server certificate (signed by your own CA) to each of your servers (and follow some kind of policy on replacing them), you DO embed a root cert of your CA into your game client (don't forget to obfuscate it! - otherwise it opens a door for self-MITM class of pseudo-attacks), and that you DO check server certificate on the client side against this stored-root-cert-of-your-own-CA. Other schemas (with a 3rd-party CA such as Verisign) are possible too, BUT they have drawbacks in gaming environment.