Main blower on all the time - Goodman GMS90703BXA Heater

No, that is a switching problem. Either a faulty switch or a glitch in the ECM from the battery being disconnected.

Sounds like your blower has a bad armature and causing to high a load on your electrical circuit. Time to change your blower motor down by the passenger feet.

Check the thermostat and make sure the fan switch is set on "Auto". If it is set on "ON" the blower will run constantly.

If you bypass tstat, meaning hot wired blower, does the blower cycle on and off by touching and removing red with fan?? If yes than obvious problem in tstat. I was leaning towards a bad relay but if you can make it work by passing tstat, there's you problem.

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Most of the time when a heater blower quits working it is the blower switch that fails rather than the motor. Blower switch can be accessed in the dash panel.

There is probably dirt blocking the main fuel supply to the engine. You need to strip and clean the carb, preferably fitting a new carb kit at the same time. Clean out all the passages and orifices by blowing them out, but be careful of loosing small parts like filter screens and screws.

Does the fan come on when you turn the Fan to ON at the thermostate? If it does then it could be the blower board not turning the low speed blower tap on. Or the Fan & Limit switch not working. If the fan doesn't come on in the on position then the blower motor could be out or the run capacitor.
If the blower comes on it the ON position then you can run the heat with the fan in the ON position till you can get the heat relay looked at.

check fuses or replace blower

it sound like the blower motor may be seizing up. try spining the blade by hand to see if it feels free or not. the motor could be trying to turn but can then tripping itself out on overload. (most motors have built in safety overloads which will trip during an overcurrant event. aka too many amps.) you may have to replace blower motor

Remove all power from the furnace, it is likely the brain in the furnace is need of being reset, from the battery replacement, why did you change the battery, was there an indicator telling you to change it?

Many times just removing all the power to the furnace will do the trick with a reset, if not let me know and we will go from there, but it sounds like the main brain is confused or bad, more likely confused, so try the power removed first.

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