The outdoor temp doesn't much affect the problem- it's the tank temperature that counts. When the tank is stone-cold (<<100F) on startup this condition is "normal" until the tank gets closer to domestic hot water temp.
If you let the air handler run until the tank drops to well under 110F it could actively drip like that. But corrosive condensation can occur at higher temps even without it dripping at the burner. Exactly where the crossover temp is will depend on the particular tank design and the pumping-rate/turbulence in the tank, the return water temps, etc. With return water at 120-125F it should be pretty safe from condensation. At 100F not so much. The delta-T on the coil is likely to be something like 20F (140F in, 120F out) or even 25F, and it may be worth bumping the temp (and installing an aquastat with ~10F of hysteresis to cut out the blower at temps in the 120-125F range, turning it on when return temps hit 130F+) if return water temps are much lower than 120. You'll have to play around with it and measure the actual temps to tweak it in.
With tight, insulated ducts and 130F entering water at the coil the exit air at the registers will still feel warm (higher than body temperature), but even with 120F water and a tepid 95F exit air it'll still be heating the house, but won't feel so cozy in the draft next to the register, but the return water temps will likely be a condensing 110F or lower when that's happening.
As a strategy to keep that from happening before adding aquastat controls, bump the tank temp up to 145-150F, and keep the room thermostats constant (no setback), so there are no long recovery ramps from setback. Odds are that most of the time with normal thermostat hyseresis it won't have to run more than 15 minutes before the thermostat turns off the air handler & pump. At 140F entering water temp the air handler is probably taking ~45-50KBTU/hr out of the tank, but the burner is only supplying (0.8 x 40K=) 32K/hr, or about a 12-15K shortfall. A 40 gallon (335lbs of water) tank dropping 20F degrees is 335 x 20 = 6700 BTUs of heat stored in the water, which at a 15KBTU/hr shortfall would take (6700/15000=) 0.45 hours, or 27 minutes.
If you add an aquastat with ~10F of hysteresis to interrupt the blower it will be cycling the air handler something like ~12 minutes on, ~5 minutes off until the thermostat is satisfied during situations like setback recovery or design-condition outdoor temps. Something like the ~ $50-60
Johnson Controls A19DAC1C available from multiple web vendors should get you there with a simple strap-on connection to the return line from the coil (no plumbing required), with current ratings appropriate for air handler motors. (Be sure to wire it for make-on-rise.)
In the longer term when it's time to replace the tank, a condensing tank with a bit more burner behind it would make the whole headache go away, and you can run the tank at a lower temp for higher condensing efficiency and lower standby loss. The 76K burner on the Vertex would deliver ~68K or more in condensing mode, whereas the original 65K burner was only delivering 52K. If you set the tank to 125-130F the return water would always be in the condensing range for 90%+ efficiency, but the exit air temps will be lower at the registers. If the exit air temp is uncomfortably low you can bump up the temp of the tank a few degrees at a time, but know that the combustion efficiency will be dropping and standby loss increasing with higher temp.