Am I right in thinking we never planned a real offensive campaign that went significantly into the north?

No, that isn't really correct. The United States, South Vietnam and allies never committed ground troops to invade and occupy the north. Much of the ground fighting took place in South Vietnam, with a few exceptions. For instance, there were some Special Forces missions that went into Cambodia and Laos to observe the Ho Chi Minh trail. There was also operation Ivory Coast which was an unsuccessful special forces raid intended to rescue POWs held in North Vietnam.

Also, in 1970 there was a "secret" invasion of Cambodia where US ground forces crossed the Cambodia-South Vietnam border to secure the southern end of the Ho Chi Minh trail and make North Vietnamese infiltration harder.

But, we cannot forget the air campaign. The United States Air Force repeatedly conducted aerial bombing campaigns in North Vietnam, including operation Rolling Thunder from 1965-68 and Operations Linebacker I and II in March and December 1972. Also, US air power put mines in Haiphong harbor in May 1972 to interrupt the flow of communist shipping bringing war materiel to North Vietnam.

These bombing campaigns were directed at transport hubs, harbors, industrial facilities, as well as along the Ho Chi Minh trail in Laos and Cambodia. The intent of these bombings was to weaken North Vietnamese ability to conduct warfare, and to "inflict pain" to influence the leadership of North Vietnam to seek a negotiated settlement ending the war that would ensure South Vietnam's independence and territorial integrity.

---

As to the underlying question of "why didn't the US, South Vietnam and allies try to invade and occupy North Vietnam with ground forces?"

There are a handful of inter-related reasons that all dissuaded US policy makers from taking that course of action.

First is the issue of escalation. Burns and Novick do a good job in the miniseries of explaining the Johnson and Nixon administrations concern about escalation, and whether US actions like aerial bombing of the North, mining harbors, invading Cambodia; would in-turn cause the Soviets and the Chinese to increase their support for North Vietnam.

You mentioned the experience of the Korean War, and a risk of repeating Chinese intervention. That likely did have an impact on US strategic thinking.

But, we should not forget the precedent of the Cuban Missile Crisis either. In October of 1962, the US threatened to invade communist Cuba to remove the threat of Soviet ballistic missiles on the island. (The failed Bay of Pigs invasion in 1961 lent credibility to the idea that the US would be willing to invade Cuba to topple the Castro regime). In response to threat of US invasion, and the risk of US seizing the soviet missiles on the island as well as cargo ships transporting missiles, the Soviets threatened retaliation. This standoff very nearly started a nuclear war.

So, shifting our attention back to Vietnam. In 1965 when US ground forces arrive in South Vietnam in large numbers, it's only a few years after the Cuban Missile Crisis, and the Soviets had shown that they were willing to take risks (putting missiles in cuba) and to escalate tensions to defend communist-aligned states. Would the new Brezhnev regime in Moscow behave differently in 1965 than Khrushchev had in 1962? Or would an invasion of the North provoke brinksmanship again?

A second factor is the domestic factor. The US government made the case to the US public (and to nations around te world) that American soldiers were fighting for the defense and freedom of South Vietnamese. And American soldiers were fighting and dying to prevent the spread of communism in Southeast Asia.

An invasion would be fundamentally in tension with the "defensive war" argument. At the outset of the invasion, the US would need to declare what their aims were. Were they trying to force the Northern govt to the negotiating table? Were they trying to force a change of regime in the north? Were they going to re-unify Vietnam under the authority of the government in Saigon?

Would American mothers and fathers, wives, sons and daughters accept troop deaths during an invasion of the North? Did the American public know or care that much about South Vietnam's independence? No matter what the US declared the invasion goals to be, there is the risk that Soviets and Chinese would read the situation as US attempting to "roll back communism", which could be expected to provoke an escalatory response. Additionally, aggressive warfare is banned by the UN convention, and so non-aligned countries around the world could be expected to take a dim view of a US invasion of North Vietnam.

Finally, to invade the North would require troops. To occupy the North while still keeping soldiers to guard the South against VC and NVA guerrilla warfare might require several hundred thousand more soldiers in-country. Rather than 550,000 US troops in the south, maybe it would require 1,000,000 troops in the South and North. Maybe more?

A million soldiers deployed to fight in Southeast Asia, in the context of the broader Cold War tensions means a lot of troops tied-down who can't respond to a potential Soviet invasion of Western Europe.

And it is tax money spent in warfare that is not being spent on Lyndon Johnson or Richard Nixon's domestic policy agenda. It's money not spent on alleviating poverty, on medicare, or not spent on infrastructure or education. Or else it is raising taxes sharply to be able to fund everything.

So, US policy-makers took all these factors into account, and decided that it was important for US credibility to defend South Vietnamese independence, but that an invasion of the North was not in US interests, and posed significant risks.