I came across those 2 sentences with söyledi.

Doktor kandaki þeker oranýnýn yüksek olduðunu söyledi. Murat Bey´in dikkat etmesini söyledi.

I want to translate this as:

The doctor told him that he had a high proportion of sugar in his blood.

He told Mr. Murat to be careful (pay attention).

So, I´m thinking söyledi with -diðini means "he said that he ...." and söyledi with -mesini means "he told him to..."

If correct, is that a good rule of thumb to use?

Yes

And this sentence:

Ancak doktorlar yine de beslenmesine ve spor yapmasýna dikkat etmesi gerektiðini söylediler.

Could it be changed to a more direct order by getting rid of gerek?

Ancak doktorlar yine de beslenmesine ve spor yapmasýna dikkat etmesini söylediler. ???

I would have let it stay there, as it tries to say, he needs to take care ...

And in this sentence it looks like the tavsiye etmek behaves like söylemek:

Doktorlar, Murat Bey´in altý ayda bir genel saðlýk kontrolünden geçmesini tavsiye ettiler.

 I think a bit stronger than söylemek, as they advise him.

So would the "rule of thumb" (if correct) apply to other words as well?

I think that´s the last sentence in the exercise I have questions about.

Thank you in advance for any explanations/corrections.

And in this sentence it looks like the tavsiye etmek behaves like söylemek:

Doktorlar, Murat Bey´in altý ayda bir genel saðlýk kontrolünden geçmesini tavsiye ettiler.

 I think a bit stronger than söylemek, as they advise him.

So would the "rule of thumb" (if correct) apply to other words as well?