What could be causing my Lennox furnace to blow cool air?
Before I get into this, I should say that if you paid a professional HVAC company to fix your furnace and they replaced the draft motor 1 month ago, you should call them and tell them they aren't finished yet.  If they told you it was fixed and it isn't then there should be a guarantee you can call them on, if there isn't than you should hire somebody else next time.  At least in the region I work, I don't know any legitimate outfits that don't guarantee what they do.  

If I understand correctly, the blower is coming on at the same time that ignition occurs, and the air that comes out of the registers feels cool instead of warm?
If this is true, I suggest:
1.  Confirm that the burners are staying lit after ignition.
2.  If you have a meat thermometer, measure the temperature of air coming out of your registers and compare it to the temperature of your house (as shown on your thermostat), this number is your temperature rise and should be at least 20 degrees Fahrenheit but ideally closer to the 35-45 range.  The manufacturers spec. for this range will be printed on the model/serial label on the inside of the unit if you want to know the exact range you should be looking for.  If you don't have a meat thermometer then find the closest register to the unit (which should have the most heat/airflow) and feel if the air is warm, you might have to wait a few minutes.  If after 5-10 minutes you aren't getting a good temperature rise (or feeling enough heat come from the the registers) then move on to 3.
3.  go back to the unit and make sure the burners are still lit.  look at the flame where it enters the heat exchanger and make sure that the flame is burning blue, is a healthy size (this will vary depending on your unit, but you should be able to tell that its getting enough gas), and isn't flickering wildly or blowing back out of the burners instead of being sucked through the heat exchanger.  If the flame looks weak then you're having gas pressure issues, it could the valve or your actual fuel coming in.  If the flame is really distorted or blasting back out of the unit then you have 1 of 2 problems:
   a. your heat exchanger is cracked/corroded or 

b. whoever installed your draft motor wired it in backwards.    

If the burners have trouble staying lit, disconnect power to the unit, turn your thermostat off, then reconnect power to the unit, then turn your thermostat back up--in that order.  Watch the unit and note the sequence of operation, and at what point the burners go out.  I can walk you through troubleshooting that if it happens.  Also look at the circuit board for a blinking diagnostic light, not all of them have one, but if yours does, it will really help you pin down the issue.  The key is usually printed on the inside of door of the unit, or somewhere easily visible on the inside.
Good luck!