Hello Robonaut, I don't know whether my coping method will work with your situation, and it's more about avoiding feelings than dealing with them, but here it is, with an example.
I personally get really anxious about wrapping things up to mail them. To my mind it equates to the thing already being broken in the mail and numerous negative feelings and self loathing follow. But I don't get anxious (as much) about the individual steps involved in the process. So l just go to the drawer and get the bubble wrap and the sticky tape. Next I just put them on the table. Then I get the item I need to wrap. Then I cut a piece of bubble wrap the right size. I just keep on step by step, trying really hard not to actually think about the process as a whole. I will still feel a little bit of stress, but at least I get it done rather than putting it off till it's too late. Perhaps this is a form of mindfulness, I don't know.
I also use a little bit of CBT to get myself to start the process at times. I did some self-help CBT for social anxiety disorder but got impatient and cut it down to asking myself just two questions - what is the worst that could realistically happen, and what is the likelihood that it will happen. I found those two questions really help me in a lot of situations. In my mail-wrapping example the worst that's likely to happen is the thing breaks in transit but it's not actually likely because I have the ability to wrap a thing up safely, and that's enough to get me going to do it.
When it comes to housework I get a sort of depressed and trapped feeling at times, feeling as if it's actually going to last forever, so these days I put on some music and think, "I'll just put the dirty dishes by the sink while I listen to the music" followed by "I'll just fill the sink with water and some washing up liquid while I listen to the music" etc till it's done. I sometimes feel I've lost time when I've done that but perhaps that's just me? Maybe set an alarm if you have something important to do later
I hope that helps. I know myself it's hard to deal with anxiety at work. Best of luck to you!