As someone who was alive and voting during Johnson's administration, and someone who was a hawk during that time, I would agree that we thought that Johnson's policies were too cautious. The thinking is pretty much the same as we hear today: if you're going to fight a war, you have to fight it to win, and that would mean throwing EVERYTHING we have at the enemy, not just "enough" to win. Many of us were convinced of the righteousness of the cause, and that righteousness lead us to believe that anything we had to do to win was OK.
Years later, with more evidence in, and a more complex view of the world, I'm not sure that I believe in the policy any more, although I do suspect that if you're not interested in finishing a war, you probably shouldn't start one.
The U.S. foreign policy known as "Containment" had grown rapidly between the end of WWII and the 1960's. Ironically, both Republican and Democratic candidates running for president in 1960 were both "cold warriors". This meant that there was little division in the early years of the 60's regarding the supression of communism throughout the world. After the Kennedy assassination, Johnson's plan was to generally follow through with Kennedy's agenda. All one has to do is read Kennedy's Inaugural Address, it is by today's standards very "hawkish". The 1964 Tonkin Gulf Resolution authorized by Congress gave President Johnson the power to "take all necessary action to protect American interests". It was the power to wage war in Vietnam, without any declaration of war from Congress. History confirms that the answer to your "true or false" question is false. Ironically, President Johnson was the president who was responsible for the greatest escalation of American troops fighting in the Vietnam War.
President Johnson is most criticized for his escalation of the Vietnam war. His policy towards Vietnam was one of escalation throughout. Whether all hawks (those who supported more involvement in the war) criticized him for not being more involved is a hard quesetion to answer because all would have differing opinions, but generally we can assume that the actions of Johnson through his administration in regards to the Vietname war were acceptable to the hawks.