Hi everyone,
I have a Ficus elastica burgundy which I bought a couple of months ago. It was happy until 3 weeks ago when I noticed some brownish spots starting on the edge of some leaves. The spots are not dry and are soft to the touch . So I thought it was caused by overwatering. Although there is good drainage and I was watering it once a week and not too much but I thought I will cut back on watering a bit. But the issue continued to persist . I have lost a couple of leaves now and other leaves are drooping and even the branches are drooping. I noticed that some of the leaves are browning where they connect to the stem and then disconnect from the stem. I have other ficus plants and they are all in the same spot and same watering schedule but those are ok.
I have posted some pictures. Any help would be much appreciated.

You have, unfortunately, done exactly the opposite of what the plant needed. That particular variety of Ficus elastica is notoriously sensitive to dry soil. It wants to be moist as much as possible and in a very bright location. It doesn't want to be soggy wet, just always moist. Most other varieties handle drying out just fine but that one doesn't. You will find it identified as Tineke as well as Burgundy.
Try to put it closer to the window and give it a good soaking. Let it dry just on the top before you water it again.

Thanks for the quick response. This ficus is 3 feet away from a large south facing window so gets a lot of indirect light but considering our recent winter weather in Vancouver we don't get much bright light these days. I have now watered the plant thoroughly and discarded the excess water. I am hoping it recovers after a while.
Any recommendations to help
With the recovery?

Warm and bright and patience.

I am extremely happy to report back that my ficus elastica is happy again thanks to John's diagnosis and his quick response! I have moved it next to a south facing window and tried to water it frequently but moderately! Looks much happier and growing again ( thank you John ).
But I have another ficus which is dropping leaves really quickly and it is not a ficus elastics. It is a ficus burgundy. I got it recently and it was quite happy but it has been losing leaves and it is now getting dark spots on the leaves . I think I should post this as a new post. Please have a look at the new post and the pictures and let me know if you have any idea what is wrong .

Your plant looks like the variegated Rubber Tree (Ficus elastica 'Tineke') rather than Rubber Tree (Ficus elastica Chroma™ Abidjan) the cultivar 'Burgundy' has non-variegated, very dark green to burgundy-black leaves.

Ficus elastica require warmth and as much bright light as possible; those with solid dark green leaves can take full sun and the variegated leaf varieties need bright but filtered light.

My thought regarding your plant is that the soil may be staying too wet with the lower light levels and temperatures that are common at this time of year. The lower light should be fine but the soil should stay just damp throughout, not soaking wet but not allowed to go completely dry at length either. Plants are usually in rest mode and not actively growing during the winter and don't need watering as often as they do during the warm months.