central A/C Turns back on 30 seconds after shutting off
Interpretation:

Ok so, the air handler inside cuts out when it should be running. These symptoms became apparent once the house was basically air-sealed with insulation. And the unit is old.

Conjecture:

Insulation is the straw that broke the camel's back (it caused the fan to have to work harder to push the air). The blower motor is tripping-out on thermal overload because it's mostly burnt out; either the windings or the bearings (or both, now) are shot.

Solution:

Replacing the blower would probably get it up and running, but do you really want to drop ~$300 + labor to find out, on a 30yo unit? (Decide soon ...winter is coming.)

Confirmation:

Turn the AC off and set the fan to 'on'. If it kicks out every half hour: there's your problem right there (provided, it's not a lose wire somewhere, but if you can set your watch to it, I'd be leaning towards an over-heat).