Do I need to run DMA
It's not mandatory for migration, but you should to avoid possible issues before you begin the migration.
Would the application face any compatibly issues or would it not as running in backward compatibility
We can not say, because we don't know the application and which T-SQL statement / features it is using.
Some old feature are deprecated, removed or have different behaviour in new version.
See e.g. Breaking Changes to Database Engine Features in SQL Server 2017 (14.x)