Hello D&M. You do not say what sort of ostomy you have and without that information it is more difficult to guess what might be the cause of the problem.
The drebris you describe is possibly the mucus created by the intestines/colon, which would normally coat the faeces to enable a smoother exit. I get it stuck to my plugs so that when I pull them out it does look just as you describe. This is normal and should not be a cause for concern.
If you have a colostomy, then the chances are that the feaces is hardening in the lower secion of the colon where the moisture is extracted. This can sometimes cause a blockage. One obvious solution is to irrigate but this isn't always possible for everyone. Another method is to drink a lot more liquids, which might help to get the output less hard. Some people use suppositories to help the passage of output.
What I would say is that one and a half months is no time at all to be getting worried about this sort of thing as it all takes time to settle down into a routine. At that stage in my stoma career, I spoke to the stoma nurse about the blockage and she showed me how to digitally extract the blockage-- not a technique that everybody would relish, but it did work until I started to irrigate.
I hope you manage to find a solution to your problem and let us all know how you get on
Best wishes
Bill