If you want a dirty white starting with a pure white undercoat isn't actually a bad way to go because you can weather it down. You actually tend to look "clean" if you start from a grey and highlight upwards, but starting from a pure white and dirtying it up can work well if that's the look you're going for.
There's various ways to weather that could be appropriate. A few ideas might be...
Acrylic washes.
Painting acrylic dark greys/browns in to crevices. Also stippling them.
Weathering powders.
Oil washes.
Unthinned oils (or only slightly thinned). You can paint them directly in to the crevices and then use a clean brush, clean tissue or cloth to blend them in to the rest of the area. The advantage of oils over acrylics for this is that you can really work them, add more, take some away, blend it in, stipple it, basically just keep working it and it won't dry out on you and leave you with an "oh crap" moment where you have to restart it because you did something that didn't work. I'd recommend using a gloss varnish before you start oiling though.
For example, this was done with oils over a pure white basecoat...
Instead of just using beige like that, you could use a mix of browns and dark greys to create a rougher, dirtier and more gritty effect.