Managing a production database environment is a delicate, sensitive, and vexing at times. I sympathize with you on the upgrade issue you had and which you so well described as I think any DBA would.
I suggest setting up a checklist that includes your daily database administration routines activities will alleviate the need to juggle competing priorities as a DBA. The checklist would typically include tasks such as reviewing server and database logs, setting up new and monitoring scheduled jobs, monitoring disk space usage, setting up and managing security, installing patches, implementing change requests, deploying builds, monitoring and troubleshooting queries performance, resolving outstanding tickets, answering routine requests, and as it may be necessary, planning for major upgrades (application, database, service packs), writing standard operating procedures and policies.
The checklist is by no means exhaustive and can be expanded upon. Once the list has been reviewed and you have secured the appropriate sign offs, I recommend sticking it on the wall in your office or cubicle right next to where it can readily be seen.
In addition, allocating a time slot to each task on the list would help pace yourself, save time, and eliminate potential errors and omissions.
I appreciate it you sharing this story.