I think they're those old horizontal slanted drop-outs.

You can move the chainring around mm's by using PROPER chainring stack spacers. You can get them in different thicknesses, down to like 0.5mm They're pretty good! You might need longer chainring bolts as well, but most likely not.

How used was each element in your drivetrain prior to assembling, and were they of different age? What amount of tension were you running? Is your chainring centered (ie no "tight" or "loose" spots in the pedal stroke)?

Once you get a perfect chainline (a shop can use a chainline guide tool to check perfectness) then get a new chainring/cog/chain all at once. Get them all in 3/32 or all in 1/8.

Beyond that, maybe it's the old frame? But I would think it's just the drivetrain.

In my opinion fixed gear bikes should be on a good, solid frame, with good solid parts all around. I would not ride a cheap fixed gear bike.
As mentioned above, a super-reliable cheap bike is an old frame, singlespeed with "set and forget for years" chain tensioner and cantilever brakes.