I'll leave it to someone else to give examples of instances where individuals were able to buy their way out of service, but I can tell you that the courts, to whom soldiers would levy suits to enjoin their deployment, recognized that the United States was engaged in a war conducted under Congressional approval. There are a multitude of court cases where soldiers with standing (as in, were about to be drafted/deployed) brought lawsuits that were summarily dismissed by the courts as they interpreted the Vietnam conflict as a sort of de facto war that had both explicit statutory and implicit Congressional authorization.

Let's start with the Gulf of Tonkin Resolution, which was the explicit statutory authority giving the Executive power to act to assist in defense of any member of the South East Asia Defense Treaty. The resolution allowed the Executive to

"to take all necessary measures to repel any armed attack against the forces of the United States and to prevent further aggression. . . [and] to take all necessary steps, including the use of armed force, to assist any member or protocol state of the Southeast Asia Collective Defense Treaty requesting assistance in defense of its freedom." Gulf of Tonkin Resolution

The resolution was also set to expire only when the President had deemed the area stable or Congress passed a resolution to terminate the resolution.

These broad powers, combined with an appropriation of $700M for the President to achieve his goals are a pretty clear green light from Congress for the President to engage in warfare. Compared to something like the 2001 AUMF that authorized fighting the Tasliban/ Al Queda, the Gulf of Tonkin Resolution is slightly more unlimited in allowing the President to determine the scope and targets of the conflict.

Now, on to the soldiers and other dissenters who challenged the authority of the president to send them abroad. We'll start with looking at Orlando v. Laird. (443 F.2d 1039 (2d Cir. 1971)). The war's been going on about 6 years, and a lot of people, including some in Congress, are starting to realize this wasn't really what they had in mind when they wrote that Resolution. Orlando, a soldier set to be deployed, sues the Secretary of Defense and Secretary of the Army to prevent his shipment overseas, arguing that such an act would only be authorized under the Constitution if Congress had taken certain actions (i.e. declared official war) to allow the President to do so. Here, the Court finds that Congress has given continued support and authorization of the war through its repeated decisions to appropriate money to fund the military, and repeated amendments to the Selective Service Act to allow for a greater number of soldiers to be drafted. The idea of continued funding of military during wartime being an express sign Congressional support for that war is somewhat contentious -- many people think that once a conflict has started Congress may be "Straightjacketed" into continued monetary support, merely out of fear for political repercussions or lost life due to troops being under-supplied. The Orlando Court, however, disagrees with this argument:

The huge appropriations annually voted to sustain the expanding combat activity cannot be read out of being as extorted by the exigencies created by presidential seizures of combat initiatives.

They believe that Congressional funding is a legitimate stamp of approval, and that just because it does not come in the form of the official declaration the framer's envisioned does not mean it is an ineffective form of authorization --- they hold that it is not up to the Court to decide/legitimize what form or manifestation of approval Congress must take, as that's a policy issue, not a judicial one.

Next we will look at the DaCosta v. Laird cases. Similar facts as above, soldier suing to prevent his deployment, but this time Congress has spent the last few years being pretty loudmouthed about their disapproval of the Executive's actions, including the decision to initiate bombing in Cambodia. There are three iterations of this case, called DaCosta I, DaCosta II, and DaCosta III, but we're only concerned with I and III. In DaCosta I, the Court says that, well, Congress is still funding the war, and the President seems to be winding down the action, so we think the Executive is still in line with Congress approval in the conflict. In DaCosta III, the President, much to the displeasure of many, has started ramping up the conflict, heavily mining several ports in North Vietnam and increasing the intensity of air and naval strikes. The same Plaintiffs now request that the Court find the President requires more approval --- he's now acting against Congressional interests by increasing his war efforts right?! Unfortunately, the DaCosta III Court refuses to consider this question at all --- these are military tactics, and the President as Commander in Chief has sole authority to make tactical decisions --- (he could be intensifying the conflict as a tactic to win and end the whole thing more quickly)-- and for the court to make any decision on whether the President was authorized to make these decisions would be in violation of the Political Question Doctrine.

Ultimately, it took a few more Court cases (See Generally Holtzman) and a multitude of statutory amendments for the conflict to die down. (Eventually an explicit decision by Congress via the Eagleton Amendment to cut of all funds for hostilities in Vietnam/Laos/Cambodia after a month pretty much ended things.) But, for a good long while there, multiple branches of Government recognized the legitimacy of the Vietnam "War," and even some soldiers who had lawyers were unable to shirk the draft orders that pulled them into the fray.

Source: Stephen Dycus, Arthur Burney, et. all, National Security Law 6th Ed., (2016).

EDIT: Formatted a Link. Somehow still didn't format it.