Tricky situation...sick leave, new offer & handing in notice
Discussion
Don't do anything until you have a proper employment contract in you hand.
Then consider timing,
When do the new company want you to start, what is your notice period, how long until you can return to action.
Triangulate these dates as best you can, and then decide what to do about handing in your notice.
Personally, I would phone my line manager when I was ready to resign, and let them know, this is the situation, they approached me before the injury, but have now made me a full offer, as I am still off sick for three weeks and my notice is four, I propose I resign now and I will return in three weeks, to complete any handover/whatever.
Don't concern yourself with how it looks, you haven't done anything wrong, just be adult about it, and have an expectation they will do the same.
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