I agree with the earlier comment around leaving as much as possible at work (shoes, for instance) and then carrying as little as possible. I use a Chrome single-strap bag but for what you’re describing a backpack would probably be more comfortable if you’re going the on-body route.
I would plan to wear bike kit unless you’re really confident you can leave early enough to never need to rush…this is an ongoing issue for me 
nothing worse than sitting down for a meeting and your clothes are sticking to you haha.
You’ve mentioned a rack a few times; a back rack will do what you describe, as would a big enough seat pack, but you should also consider a front rack or basket. You can get a nice wide load in there without it interfering with your legs like a back rack, and if you’re planning to get a new bike anyway, you can find one with the mid-fork mounting points for a front rack.
Path Less Pedaled on YT has a deep list of reviews for the type of stuff you’re looking for, too!