In key signature Eb maj (4/4 time signature) I have a quarter note Eb on last beat tied across the bar (where the key signature changes to E maj) to strictly a D#. However as soon as I make the tie Dorico switches the D# to an Eb. Other than using a slur - is there a way to have Dorico “know” the notes are the same sound, but respect the key signature change and so show D# when using a tie?
Try creating a second voice (if one does not already exist on that staff) and putting the D sharp in that voice. Then select both notes and press T to tie them. Because one note will be on a line and the other on a space, the tie will not be horizontal. To avoid potential problems with following notes being in the wrong voice, it might be a good idea to input some notes after the D sharp, but in the original voice, before changing the voice of the D sharp. You will probably also need to 1) select the D sharp and enable Ends voice in Properties and 2) select the first voice’s rest and do Edit > Remove Rests. If the tie curves the wrong way, it can be flipped in Engrave mode.
There is no need for a second voice. You can switch to engrave mode, select the tied-to Eb and hold down Alt/Opt while pressing the hyphen key to change the tied-to note to a D#:
I tried your solution with a c sharp tied to a (former) d flat, but when I press Alt+hyphen, Dorico changes the second note to a b double sharp, not to a d flat. Do you have any idea about this behaviour?