Question on driving licences in the EU

Have a question about driving licences in EU countries.. If an EU citizen moves his permanent residence to another EU country, he must then also make a new driving licence? For a few years I have been representing our company in the Czech Republic and have my residence here. Yesterday evening a legal guardian explained my German driving licence is not valid. According to EU law I have to apply for a new driving licence here!! To conclude the contract then the request, i I’m supposed to leave my car and walk home, after all, I’m not in possession of a valid driving license. Is anyone familiar with the legal situation or has had similar experiences? Best regards Volvo 174