This is a good example of the limitations of diagnosis by internet.

We have words, but no context, so have no real idea of what's going on. For example the OP initially described the rim as having large spoke offset. Then when there's a picture, i turns out to be a (sort of) Westwood pattern rim, with not much offset at all relative to the width.

We also don't know the rim's construction, ie. single layer with rolled sides, or formed with full overlap at the center, so to layers thick at the center. Knowing the construction is important because it helps predict rigidity. However Westwood rims are fairly rigid as steel rims go., though of course that depends on the strip gauge.

Then we don't know what kind of tension we're taking about. The OP has described the tension, but we can't know his frame of reference. It's like a doctor asking a patient about pain, using a scale 1-10. The problem is that one person's 3 is another person's 7

Then, while the OP describes the wheel going out of true, we don't know the extent, whether it's wobble, radial, ot twist (one side rises and falls out of phase with the other, which is common with single wall rims.

Lastly, we don't know technique. Is the OP adding tension, one flange at a time (alternating spokes) or spokes in sequence? Adding tension by full, half or quarter turns? How about sticky threads, is he correcting for twist as he goes along or not? If so, how? I see real possibilities that sticky threads and twist mean that the OP is introducing the problem via his technique.

I'm not saying this is the OP's fault, or exactly what's going on, simply pointing out that any analysis cannot be better than the quality of the information given. Or, as they say in the computer world, GiGo.

BTW - for the OP. This may be an example of why it's sometimes better to leave bad enough alone. The spokes "seemed" loose, but the wheel was true, and given it's age, we can safely assume it's been that way for a long time, and would probably stay that a while with light use. If not, there's always time to fix it AFTER a problem arises.