I have got SQL Server Desktop Engine running with 2 database and I need
to install the evaluation version of SQL Server 2005 instead. A
straight upgrade does not seem to be possible. Does someone know if
backing up the databases, uninstalling the desktop engine, installing
SQL Server 2005 and then importing/restoring the databases would be an
option to look into

Converting Desktop Engine to SQL Server 2005
S4lbullet
Here is an article on Upgradeing MSDE to SQL 2005 Express, you could follow this example then it is an easy upgrade to the Full SQL 2005 Product.