Two techs, two answers on my rental furnace, how do I get this sorted?

The test that settles this is a combustion analysis, done with the furnace running and then again right after the blower kicks on. If the CO or oxygen readings shift when the blower starts, that's air from the duct side getting into the combustion side, wich means the exchanger is compromised. Just waving a CO meter at a supply register tells you very little. I've seen cracked exchangers that never pushed measurable CO into the ducts because of how the air pressure balances inside the cabinet. Zero ppm at the vent after ten minutes proves almost nothing on its own.

The other cheap check is the pressure switch situation itself. On an 80% furnace, a blower running against a breached exchanger changes the airflow the pressure switch sees. If that thing keeps tripping or trips intermittently, that's not a sensor problem to swap parts at, that's a symptom.

On the landlord side, I went through almost this exact standoff in a rental a few years back. What finally moved him was a written report on company letterhead from an independent tech stating the unit was unsafe to operate, plus a copy to the gas utility. When the utility's inspector showed up and pulled the meter themselves, that was the end of the argument. A red tag from the utility carries legal weight in most places that a contractor's oppinion doesn't. A city or county housing inspector can also work, since many jurisdictions treat no heat in winter as a habitability violation with a tight response window.

Keep every written estimate and report you've got, and send the landlord one dated email summarizing the situation so there's a paper trail. If he refuses and you document that you notified him in writing, you're in a much stronger spot, and in a lot of states you can pay for the repair and deduct it from rent, though check your local tenant laws before doing that.

Space heaters in the meantime, keep them well away from anything flammable and don't run extension cords to them.

Worth calling in a third tech yourself, one who's got no dog in the fight. When I had a similar standoff with my old place, paying out of pocket for an independent inspection was like 150 bucks and it ended the back and forth. Two written opinions saying the same thing is hard for a landlord to wave off. Also, frame it his way. A cracked exchanger on a 14 year old unit isn't a repair, it's a replacement, and if something happens after he's been told in writing, that's on him. Landlords move quick when liability shows up in an email.

Been chasing cracked exchangers since before camera scopes were a thing. Old trick my mentor taught me: pull the front panel, watch the burner flames with the furnace running, then let the blower kick in. If those flames dance, stretch, or roll out of the burners when the blower starts, air is blowing where it shouldn't be. Takes two minutes and no fancy gear. And that sensor swap the landlord's guy did smells like a band aid to me. Pressure switch trips on an old crusty unit are ussually a symptom, not the disease. When I was renting, what moved my landlord was asking his own tech to sign a note saying the unit was safe to operate as-is. Suddenly the story changed.

One thing I'd push back on from the landlord's tech: surface rust and an actual crack are not mutually exclusive, and you can't judge either one through a layer of scale. A proper scope inspection means brushing the flue passages clean first, because rust flakes off in sheets down there and hides the corners of the cells where cracks usually start. Did the first tech give you photos? A picture showing the exact cell and where it's split is about the hardest thing for a second opinion to argue with, and if he shut the gas off he should have documented why in writing.

Update: we got the independent third opinion like a couple of you suggested and paid for it ourselves. He ran the furnace with the blower off and then on, and the readings jumped both times. Cracked for sure. He also said the flames flickering when the blower kicks in is the old-school tell, and ours did it. Handed both reports to the landlord and the replacement is scheduled for Friday.