I've been watching the news, and I've/I heard that

[EngLearner]

I made up the following sentences:

1. I've been watching the news, and I've heard that President Trump is going to offer a peace deal to end the war between Russia and Ukraine.

Context for version #1: I've recently gotten into the habit of watching the news. Say today is Saturday, and I've watched all the news broadcasts since Monday. Several times since then, I've heard the news anchors talk about Trump and his peace deal.


2. I've been watching the news, and I heard that President Trump is going to offer a peace deal to end the war between Russia and Ukraine.

Context for version #2: I've recently gotten into the habit of watching the news. Say today is Saturday, and I've watched all the news broadcasts since Monday. In today's news broadcast, a news anchor mentioned Trump and his peace deal for the first time. By the simple past "I heard", I'm referring to that specific news broadcast only.

Question:

Can sentences #1 and #2 be used in the contexts I've provided? Do you think my intuition about the bolded tenses is right?

 

[emsr2d2]

Note that "I've been watching the news" without a timeframe suggests that the watching happened in the very recent past (like in the last hour or similar). You need to be clearer if you mean you've recently been watching the news regularly. Here's how I would say what I think you're trying to say:
I've been watching the news a lot over the last week and I keep hearing about Trump's "peace deal".

I realise that that doesn't help you to choose between "I heard" and "I've heard" but I would use neither in your original sentence. If I had to pick one, it would be the present perfect.
I've been watching the news regularly over the last week and I've heard a lot about Trump's "peace deal".


If you want to express the regularity with which you've heard information about that subject, I'd use the continuous with the verb "hear".
Recently, I've been hearing a lot on the news about Trump's "peace deal".

As far as expressing that you heard a piece of news for the first time in a specific news broadcast, I don't see the need for you to explain that you've watched other broadcasts recently.
On the news today, I heard about Trump's "peace deal" for the first time.

 

[Piscean]

You are trying to pack too much information into use of tense and aspect, @EngLearner. Note how much more clearly the intended meaning comes through in emsr2d2's suggestions.