yes, if you meet all the requirements for being eligbile for dagpenge - then it is no problem that you already have a job-contract.
However, one of the requirements will be that you are willing to search for (another) job and go to meetings in the Jobcenter and in your A-kasse.
In other words you must be "available for the labour market" - this also means that you can not go on holiday in July and August, and at the same time receive dagpenge.
Finally you must be aware that for being eligible for dagpenge, you must also meet the residence requirement. In 2019 the requirement is 5 out of the last 12 years. You tell you have stayed in Denmark "approximately 5 years".
This residence requirement must be met on the day you apply for dagpenge.
If you meet all the requirements, then you must be aware that if you have not been (student-) member in A-kasse, for at least 1 year when you graduate, then you must accept 1 month quarantine/suspension period, before you can begin to receive dagpenge.
Regardless of you will be able to get dagpenge in July and August or not, then I strongly recommend you to become member (if you are not already) at LATEST 14 DAYS after graduation. If you do that, then you will have the right to start to receive dagpenge up to 2 years without needing to meet further requirements.
Let me know if you have further questions.
Anders Weber, www.a-kasser.dk