How do I control which text links have an underline applied?
By default all text links get exported with an underline.
The global setting for this can be changed by going to Utilities > Web Properties > Website tab and unticking the option 'Underline Text Links'
Note that this controls the behaviour for all text links in your website file.
If the link is being applied to the whole text object then by grouping it to itself will cause the text to be exported as a graphic.
Select the text object and go to Arrange > Group.
Applying a link to a single word or phrase within a text area
If the link is applied to only some of the words in the text object then;
Go to Utilities -> Web Properties -> Website tab and untick the option 'Underline Text Links' which will remove the underline from all links in your website document.
For text links that you want the underline to be applied to, select the text and apply the underline as a text formatting option using the Text Tool.
Then right click on the selected text and go to Web Properties -> Link tab and untick the option 'Use common website link colours' and click OK.