Your translation is spot on.
There is absolutely no problem ending a sentence with a verb in the te-form. It is done quite commonly. This is, of course, more direct than adding ください or ね to the ending, so in conversation it has the possibility of sounding pushy, but not necessarily in a broadcast, etc.. If you are addressing someone in a higher social position who you should show deference and politeness to, this type of directness should probably be avoided.
Polite and
friendly--------------------Polite-------------------------Friendly----------------Direct
気をつけてくださいね ⇒ 気をつけてください ⇒ 気をつけてね ⇒ 気をつけて