How do pilots/controllers deal with overlapping transmissions on the radio?
Well in regard to "How are these situations resolved?" you've already answered your own question in a way: the intended recipient responds with something like "Blocked" or "Calling center, say again." (If they are expecting to hear from a specific aircraft they'll often respond with "N12345, say again.", and if the frequency congestion is really bad controllers may add "All other aircraft standby").
As kevin42 pointed out this is very similar to what digital CSMA/CD systems do - they detect a collision ("Blocked"), wait a while, and try their transmission again.



As a practical matter, two-way voice communication over the radio is possible, but difficult: for example a discrete "uplink" and "downlink" frequency as you have with some radio systems would be impractical in the sort of mesh topology that you have with aviation radio (you could implement it for a single air-to-ground pair trivially, but aviation radios are also used for air-to-air communication, and it's important that pilots hear what's going on as this enables them to build a mental picture of the airspace, as well as ensure that they're not blocking other pilot's transmissions).

Regarding analog-vs-digital, the traditional AM radio used in aircraft has one major advantage over its digital counterparts: when two digital signals overlap the result is garbage -- you can't recover any of the information. When two analog AM signals overlap you get heterodyning (the two signals mix, and your receiver pipes that mix into your ears).
The resulting mixed signal sounds AWFUL, there's an ear-bleeding squeal from the out-of-phase carriers and the voice modulations combine to a garbled mess much of the time, but with experience you can usually pick out one of the communications (particularly if you're close to one of the transmitters, where it will overpower the more distant signal).

The ability to step on a transmission as described above this can be useful: Instructions have been issued over a "stuck mic" before to get other aircraft off the frequency.

In a similar vein, if you have an emergency you probably aren't going to listen before you talk - you're going to key the mic with your mayday and even if the controller misses half of what you're saying under the heterodyne noise they'll know there's an emergency when they hear the other half of your transmission and tell everyone else to standby while they talk to you.  

In both of those situations a digital system implementing CSMA/CD would detect that the frequency was in use and not transmit (and if were a digital system and transmitted anyway both digital signals would be garbled and unusable).