Hi Drew!
We all have different solutions, but I carry two changes of diapers, travel size adult wipes, disposal bags, small amount of extra tape, small tube of barrier cream, and latex gloves in either a small backpack or a military style messenger bag. I think the latter is slightly less conspicuous when going into a public restroom.

For work, a top-opening briefcase is o.k., just make sure that the compartment with your diapers is closed or put the contents inside an opaque plastic bag.
I would not say that I have learned not to care, but, over time, to have it bother me less.
In my car I keep a more complete "disaster kit" with more of the above - plus a full change of clothing in case I have soiled my outer clothing. I am double incontinent and worry most about a fecal accident.

There is an extra plastic bag for wet or soiled clothing. I also have a disposable absorbent pad to protect the car seat. This kit does is not often used, but may provide more peace of mind just having it, just in case.
I change wherever I can - a disabled or family bathroom is preferred, otherwise whatever offers the likelihood of a relatively clean stall and at least some minimum privacy. Sometimes we cannot be choosy.
Although immediately disposing of a used nappy is preferred, I always am prepared to carry away a used nappy (carefully sealed in a disposal bag) in my pack for disposal elsewhere.
If you have to change while on-the-go, consider changing earlier rather than later, particularly if you have found an acceptably good place to change.
Drew, we all have been in your shoes. Good luck.
--John