How do I continue to work in close proximity, whilst there is this uncomfortable tension?
First, you'll have to accept that things will be uncomfortable for a while. This will be harder if you're new because your co-worker won't know whether this is out of character for you or not. There really isn't anything you can do about it directly beyond the apology. If you behave normally and professionally, you should find that you can continue to work together even if it doesn't feel great for a while.
The best apologies:
- Admit wrongdoing.
- Are clear about what you're apologizing for.
- Include a promise not to repeat the offense.
- Include an explanation of how you plan not to repeat the offense.
(When apologizing to someone you're going to continue to come into contact with.)
You're apologizing for both not following procedures and for being dishonest about the extent of it. You can address not following procedures by following them from now on. It'll be important to do this even when you feel an exception is justified if you don't want people to think your apology is insincere. (Later, when you've got some trust back, you might be able to ask for an exception to the rules ahead of time but you'll probably be better off explaining the problem and asking for help than proposing going against procedure up front.)
You address dishonestly by being honest from now on. If you can find ways to be more transparent about what you're doing, that may help, but proving you're honest just takes time.