Most Christmas markets do not require tickets. There are some exceptions - like markers held at castles.
Yes, it is more difficult to find lodging during the markets, and much more expensive. We stayed in Colmar in June and we are returning for the markets in December. The price at the place we stayed more than doubled, so we are staying in Strasbourg (bigger city, more options) and taking the train. Strasbourg is also known for its markets. There are many towns nearby that have markets also.
The timing of your visit is important. Many markets open around 11/27. Many markets in Germany close on 12/23 or 12/24. Some markets in Austria are open until 12/31. You need to look at the official website for each market to be sure.
We are flying into Frankfurt, spending one night then taking the train to Strasbourg for 4 nights, then Cologne for 3 nights. Each city has several markets, plus many more within an hour by train.
Salzburg would be lovely during the markets. It’s a gorgeous city with tons to see any time of year. You could easily combine it with Munich.