Integration courses are offered at every adult education centre [Volkshochschule] in Germany. Many also take place online. You will find information about this here.
You will also find information on the websites of the respective federal state and social welfare offices. Here for Leipzig, for example.
An example of an integration course with childcare can be found here.
You will find language courses specifically for Ukrainians from the Mitteldeutsches Institut (MIQR) here.
In the context of adult education, language and integration courses are also offered, for example, by LBW, WBS, inlingua, Sprachinstitut Leipzig, Sächsische Lehmbaugruppe and Euro-Schulen.
As you can see, there are many different providers. Use BAMF-NAvI for more information.
Most providers will help you fill in the application form for BAMF. It is best to ask for this help in person.