Dear @Simi,
Welcome back to the Wefugees Community and thank you for being such an active member.
Making an "Ausbildung" is a very good idea and in general, it is also possible to do it part-time. For this, you have to find an employer/training company (you want to make a dual vocational training including working with an employer and going to school, right?) which agrees on the part-time model. However, I have some doubts if changing the employer and school after a few months again (in case you move to another place) makes much sense. Do you know already where you are going to move? If yes, I may recommend to apply for an "Ausbildung" at the new place since the training programmes ususally start in August/September and now would be the time to apply respectively. An Ausbildung usually requires a German level of minimum B1 - you could use the free time to work on this as well.
In case you want to stay at your current location, an Ausbildung might be a reason not to be transferred to another place. I highly recommend to discuss your best options individually with a migration/refugee counselling office.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike