It sounds to me like your 'safelight' isn't so safe... You say it's a "a white lamp with a 15W red lamp"; does that mean you're using a white bulb and a red bulb together; a white bulb inside a red lamp housing/shade; or a red bulb inside a white lamp housing/shade?
Black and white papers (except Kodak Panalure (old) and the Ilford Digital range (current, only supplied in rolls)) are sensitive to green and blue light, which is the reason safelights are red, orange or sometimes yellow. You need to eliminate *every* source of 'white' light - red-coloured tungsten bulbs still emanate a percentage of blue and green light to which your paper is sensitive.
On the other hand, your paper may be completely fogged, in which case it's useless for its intended purpose. Here's how you can test it. In complete darkness, develop and fix a sheet from the middle of the paper packet, and one from the top. If they're large sheets just cut a strip 2" wide along the top. If you see any greyness after fixing, the paper is fogged.
If your first test comes out white, take another sheet or strip in complete darkness, lay opaque items like coins on top and switch on your safelight for two minutes. Again in darkness develop and fix this sheet or strip; if the outlines of the items are visible, your safelight is unsafe. If nothing is visible, try the same test using the same methodology for five, ten, twenty and forty minutes. If there's no visible fogging after forty minutes under the safelight, that's safe enough for everyday darkroom work.
At a pinch, some red ultrabright LEDs wired to a battery and wrapped in red polythene will be safer than a red tungsten bulb.