I've had a feeling of tightness in my chest for over a month now. It is most bothersome at night as I can't stop coughing. My cough is most often dry but on occasion I do cough up small amounts of mucus.
A little history about me is that I am a full-time welder and smoke about a pack of cigarettes a day. I'm wondering if I'm having early signs of asthma as this coughing and feeling of tightness in my chest has been consistent for over a month now.

While it could be, at least with my asthma there is normally a trigger that makes it worst, a cat, smoke (the smoke in your house if you smoke in it all the time could make me so sick I would have to get out of it) Not every one is triggered the same.

You did not mention how old you are, how long you smoked, you blood pressure levels, and other issues that can cause various issues.

If you are on blood pressure or other medicine for a long time some people start getting a reation to some brands and it caused coughing.

Also I woul be more concerned over possible gases and metals that could be inhaled at a welding shop.

My best advice is that if it is astman it needs to be looked at and proper medicine taken, If it is not, you need to find out what.

Go to the doctor, get a chest X-ray and breathing tests.

Hi, welder,
Have you seen a Doctor?
If anyone, regardless of what occupation they have, developes a cough and it persists, they really need to see a Doctor. A simple X-Ray might reveal the problem, might not. That is up to the Doctor.
I don't want to alarm you, but it could be a sign of Lung Cancer. That is why persistent coughs should be explained to a Doctor. Hopefully, it will not be something that serious. I do wish you the best, and good luck.