There are a couple of things that are strange here. First, you have low expression in the presence of galactose. The GAL1 promoter is really strong, so that strikes me as odd. Even if it was leaky, induced vs uninduced should be night and day. However, you should keep in mind you still may not be able to see it by eye. Second, it is weird that it would be leaky in the presence of glucose (or dextrose, in your case). The glucose should not only prevent the promoter from activating, but it should actually repress it.
A few potential fixes that come to mind:
1) Check your strain genotype. Some strains do not have gal4, the transcriptional activator of the GAL1 promoter, so if you don't have this, the system will not work in your strain. Likewise, you may not have an intact UAS, which would make the repression fail. Other issues may also arise which may make a strain unsuited for this system.
2) Sequence your promoter/gene. It is kind of a pain, but if you have a mutation in your Gal1 promoter, this could be the reason for all your problems. Also check the UAS, as this would create repression problems. Finally, check if the gene itself is inserted in the place you think it should be, if you got some recombination mishap this may be the problem. The appearance of an extra band at a low molecular weight makes me suspect this may be an issue.
3) On that note, try picking a different clone. Maybe you got unlucky and the colony you picked had a defect, and a different colony will be what you think it is.
4) If you haven't already, make fresh stocks of your galactose and glucose media. Maybe you got impurities or some other issue in your media, and therefore what you think you are giving the cells is not what you are giving the cells.
Anyway, that's all I got. Good luck!