To converse requires being up to speed, it takes time to get up to speed and in that time the person contacting you will grow bored, so the only solution is to be at full speed all the time (see my overclocking essay). Being at full speed all the time causes stress.
For some people, conversation comes naturally and does not require lengthy though. You can find many of these people on IRC, or in AOL chat rooms :-P. For these people, having email and IM always on is not a problem as they do not have to be overclocked to participate.
I am not one of those people. I no longer use IM, and I use fetchmail to get my mail -- I don't even have the option of getting alerted every time a new mail arrives. I've had this setup for a while, but it's taken me up until now to understand why :-).
The solution to stressful IM/email is pretty simple once related to overclocking: slow things down, so that you can not respond with great speed.
Set up your email reader so that you can't reply to an email until, say, half a minute after you have read it fully, and so that when you send an email it is held for fifteen minutes before being sent (in which time you can cancel it).
Set up your instant messager so that incoming messages are held for several seconds before being displayed (possibly for a slightly random period), and that you can not compose a reply until several seconds later, and that the sending of that reply is delayed (and can be cancelled during that delay).
This would also serve to reduce hot-headed replies that one later regrets, and generally stimulate a more intellectual and less, er, IRC-ish style of conversation.
The level of delay you use might even be variable and visible -- other users could see how real-time your were feeling. To politely log off, you would gradually increase your ping time (leaving IM conversations is something I always feel awkward about).
Alternatively, I could discipline myself not to reply instantly, but that will take practice.