it is only free as long as you are studying. So if you want to keep A-kasse membership after graduation, you will need to pay membership fee. However if you work maximum 30 hours per week, you can have a part-time membership. By this you will save DKK 118 per month (in 2019) compared to a full time membership.
With a part-time membership, You will still be able to get paid dagpenge as a full time insured, if you just are willing to be available full time for the labor market, i.e. seek full-time job when you are unemployed.
Be aware that the residence requirement probably will be canceled again. The new Government in Denmark have said that they want to remove the residence requirement. It will probably be removed in 2020. But off course we can not guarantee what the Government will do or not.
I would recommend you to get registered as an graduate in the A-kasse. After that you will have up to two years where you can apply for dagpenge based on your graduation/education. So if the residence requirement will be removed, you could get dagpenge in 2020.
With a part-time job, you also have the option (assumed that the residence requirement is removed) to apply for supplementary benefits.
So if you want to keep/pay for membership or not, is up to you I think. But now you know a little about your options.
Anders Weber, www.a-kasser.dk