My question to the OP is, exactly what is your CFI expecting from you? A diversion is simply changing your route of flight in response to an event, whether it be a mechanical problem, deteriorating weather, running lower on fuel than expected, or needing to use the restroom. The =goal= is to get from here to there efficiently, not doing a bunch of calculations.
OTOH, some CFIs feel the need to inundate their students with a bunch of extra make-work. I guess it has its place since (as you found) it provides a distraction and requires you to learn to juggle multiple tasks. But if a CFI uses it for this, the training purpose should be clearly understood and differentiated from the real world.
My approach to teaching it is pretty similar to what Speed Pilot described. As some know, I can't stand mnemonics, but managed to come up with one for diversions in response to an online challenge a few years ago:
The 4 F's
Find it - select the place to divert to
Figure it - estimate the direction that you'll need to fly to get there.
Fly it - Get going!
Fine tune it - now that you're heading there generally, take the time to make sure that your estimated heading is correct. This may be tuning in an on or close to field navaid, the "roll the pencil to VOR" method of determining magnetic course, rules of "thumb" about distances, or just locating landmarks that tell you you're on your way.
(someone once pointed out there's a 5th "F-it" - what you yell when the emergency actual happens
)
I also follow John King's advice to land at an unplanned airport on dual cross countries. I usually plan an in-flight "problem" ranging from bad weather to a system failure during a dual cross country.