So, now that I've gotten a new gasket for the master cylinder, it helped a bit. I can pump the brakes to where they're tight but it fairly quickly loses pressure again.

Is it just 'air in the lines' or could it be something worse? I don't have time to bleed them today but I will as soon as I get the chance.

I really think you need a new master cylinder. The reservoir chamber is not meant to pressurise when you step on the brake, so the cap gasket leak was not really causing your lack of brake system pressure. With the cap off, the only time you should see brake fluid spurting up in the master is when you take your foot off the brake pedal. No pedal + spurting fluid when you apply the brakes = master cylinder leaking internally.

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