I have had shorness of breath on synthroid. I am still going. It takes a long time before you see any improvement. Make sure you have your labs done at six weeks to make sure your optimal.
I have had tightness in the chest but not from thyroxin, it was from being Hypo.
Keep an eye on all your symptoms and the advice I would give is to have an ECG done to make sure all is well.
Your local Doctor can organise this.
Personally I think its the Hypo causing those symptoms but best to err on the side of caution.