whether you can keep membership of a danish A-kasse when getting a job in another country, depends on which country it is that you will have to work in.
If you get a job in another EU/EEA country then you need to cancel your membership in the danish A-kasse, and get insured with your new EU-country's unemployment Insurance Scheme. Remember that you must be insured in your new EU-country within 8 weeks after you have ceased to be insured in the danish A-kasse.
If your new job is in a non-EU/EEA country then you can keep your membership with the danish A-kasse.
When you have a job-contract for a new job, you do not need to search for job the last 6 weeks up to the day you have to start the new job/stops receiving dagpenge.
The A-kasse will tell what kind of proof they need, but a job-contract will be fine I think.
Anders Weber, www.a-kasser.dk