Methinks you are using a hammer to cut a piece of glass in two. Certainly, after using a hammer, the glass WILL be in at least two pieces, but perhaps not in the shape you originally intended. That is, the Law of Unintended Consequences and the Law of Indiscriminate Application of Force (non-controlled application of forces results in non-controlled consequences).
I suggest that, rather than a law, you should make it compulsory for all politicians or candidates for a public office be required to take an oath, much like one does in a court of law, to always tell the truth in any communication made in association with that office or campaign for that office. Thus, if they did purposefully tell a lie, the offense would not be in telling the lie, but in breaking their oath. Breaking their oath would be grounds for their immediate removal from office, at a minimum. As part of this oath, they would commit to always support any communication they made with well documented references to support their communication, much like companies are required to back up any claims they make in an advertisement with proof. Failure to make public this evidence would mean either a public retraction or removal from office.
The difference between breaking an oath, and breaking the law, is in who enforces the infraction? If one is accused of breaking a law, it means charges are laid, it goes before a judge, and you have the entire legal hassle of 'political immunity from prosecution', legal appeals, legal procedures, and the adversarial process of courts.
If, on the other hand, it is an oath, then an entirely different process would be followed. It is no longer a criminal offence. It would come before a tribunal (an impartial, non-partisan jury of peers, perhaps) that would hear the accusation, hear the evidence, determine the truthfulness of the statement, and make a binding decision as to weather it violated the oath or not. Sort of like the advertising council tribunals. The procedures would not be the same as used in a court of law. It would not necessarily be prosecution-defense-rebuttal-rebuttal, discovery, and the prosecution laying the case out at the beginning. More along the lines of an inquiry, where dispositions are heard, evidence presented, and a decision made.
Another difference is that being accused of breaking the law means everything has to be spelled out, clear-cut, unambiguous, and specific. Breaking an oath, on the other hand, is more along the lines of a civil, instead of a criminal, case. 'Reasonable probability' is substituted for 'beyond all reasonable doubt'.
In such a scenario, it would be up to the tribunal to decide if the communication were made as something that the public would take as a statement of fact and rely upon in making decisions. Very similar to how advertising councils and newspaper tribunals make decisions. And just as in such existing tribunals, allowances would be made for subjective 'fluff' through common law and precedent. Only claims that could be factually analysed and objectively verified would be subject to the oath. 'Largest carrier by installed base' is an advertising claim that can be supported or refuted. 'The greatest thing since sliced bread' is entirely subjective, and therefore can not be supported or refuted.
Since advertising tribunals do work to cut down on deceptive advertising, there is clear evidence that such a system could work, as long as the tribunal had teeth (removal from office and sanctions against participating in campaigning again).
As much as you wish to stay away from the current political situation in America, I posit that it does provide substantial test situations in which to evaluate the effectiveness of whatever method you use, and whatever wording you use. Pick any or several of the hundreds of examples of situations in recent American politics that you wish to address, apply your solution to the scenario, and see if it results in the intended consequences.
Could a politician wiggle out as easily as they can currently from the consequences of statements they have made? Does it cast too wide a net, and implicate politicians in situations you feel they should not be admonished for?
For an example of what an oath would look like, consider the Canadian Code of Advertising
As its name implies, the Code has as its primary purpose the
expression of Canadian standards in advertising that, when followed,
should result in responsible yet effective advertising without
unreasonably blunting the underlying fundamental right to advertise
lawfully-sold products and services in a fair but competitive manner.
The oath would state 'I will not violate the 'Political Code for Truthfulness', and the code would spell out the criteria and determinants for what is a false or true statement, much like the Canadian Code for Advertising. This Code would be flexible and interpretable, a guideline for the tribunal to use, just as the Canadian Code to Advertising is intentionally somewhat malleable.
As an adjunct, the Advertising Code contains a provision for determining the legitimacy of a complaint
If, upon review, it appears to ASC or Council that a complaint is not
a disguised trade complaint or special interest group complaint, and
that based on the provisions of the Code reasonable grounds for the
complaint appear to exist, then the consumer complaint will be
accepted for processing. If at any time thereafter during the
complaint review process, but prior to the release of Council’s
decision on the complaint, either ASC or Council concludes that, in
reality, the complaint is a trade complaint or a special interest
group complaint, but not a consumer complaint, the process will be
discontinued and the complainant notified accordingly. In these cases,
the complainant will be reminded that alternative approaches should be
considered by the complainant for registering an advertising-related
complaint, such as under ASC’s Advertising Dispute Procedure or
Special Interest Group Complaint Procedure.
But trying to spell out the Code in a simple 'Law' format is nigh on impossible. There are so many intricacies, that it would be impossible to state clearly and unambiguously.