Methotrexate and persistent cough

Hi everyone. I was diagnosed with what appears to be psoriatic arthritis in October 2020. Although my condition could be rheumatoid arthritis, my nails are flaky and this is a sign of psoriatic arthritis.

I took Sulfasalazine for 3 months but no real improvement and then the last 2-3 months, I have been taking Methotrexate. My condition greatly improved and inflammation in my feet and hands went down. However, I developed a persistent, somewhat violent, dry cough and so my doctor as taken me off Methotrexate. So now I am only taking Sulfsalazine but the pain in my feet, hands and knees have come back quite badly. This week has been painful. I have the following questions if anyone could help me:

1) Is there a combination of drugs that could help me in my condition but not have the side effect of the cough?

2) What if I take Methotrexate in the long-term? Could my lungs be damaged?

3) Could anyone suggest alternative remedies such as homeopathy or natural medicine to reduce inflammation? (I take cod liver oil and ginger tablets everyday and am avoiding gluten but something stronger)

4) I have heard of this US medicine called 'Heal n Soothe' (https://healnsoothe.com/). Does anyone have experience of this and could they recommend it?

Thanks for your help.

God bless you.

Imranali