it seems that the initial surface has some issues on that specific edge. i believe it was made from 2 curves which creates a singularity on the tip. the only way to fix this is to create a non singular surface with patch for example.
since i hardly can see anything from your surface this is a blind suggestion. but you can extract the isocurves from the original surface with direction both and option extract all then delete the surface join the boundary curves if they are split then use patch on all curves with something like stiffness 0.5 and 20 spans in each direction with the option automatic trim. you may have to experiment with this if you need a joinable edge from the same boundary. anyway this will create a blanket which will be trimmed off around which you then can offset without problems.
if it does not work out you may post your file.
@RichardZ Thank you so much for your reply… yes patch is a good alteration but anytime I sweep2 between 2 curves and then offset I have this problem.