I see what you mean. A news website is excellent for practising listening but as you point out, specific activities need to be designed when using transcripts.
First of all, is there anyway that you can cover the screen while students are listening? You could put a piece of paper over the computer screen or even switch the screen off (of course, it’s important that students can still hear).
If you decide to let them see the transcript, listening and looking at a transcript is an excellent way to study pronunciation. You can take a short section of the recording (two or three sentences) and ask students to mark pronunciation on the transcript. You can either print out the transcript or ask students to copy it. They can then mark intonation, stress and unstressed syllables on the text. They can also look at which words are linked and which sounds disappear in connected speech. After marking all this on the transcripts, they can practise saying it and compare their speech with the recording on the news website.
For extensive listening, it is usual to focus on listening for gist. This means getting the general meaning rather than a series of questions that look at specific information.
With a news story, this could include questions such as:
What were the main news items? War, earthquake, other natural disasters, happy stories, etc.
How many news items were there?
Did the events involve individuals? Governments? Nations?
For students at a higher level, they could write a short summary of one or two news stories after listening. They could also discuss the news items they heard. To make the writing and speaking activities more communicative, half the members of the class listen to one story and the other half listen to another story. They then tell each other what they have heard or read each others’ summaries. This could also be followed up by each group listening to the news story that was originally played to the other group.
I hope you and your students will enjoy these activities. Good luck.