Yes you are quite right, 'dig' here means to like or to enjoy. It is very popular in modern slang. I definitely hear it more in US English even though I know the Housemartins are British.
We can even use dig in present continuous:
'I'm digging your hat' - I really like your hat
Of course it is only used very informally.
With the models line, it is possible that it is supposed to be ambiguous. As with many song lyrics, there is often more than one meaning, and of course sometimes the meaning isn't even clear. For that line both models as in fashion models and models as in race car models are possible for both of the words. Possibly they are talking about fashion models with brains the size of (race car) type models. Of course a (race car style) model could be any size, so the meaning is still ambiguous even if you know the meaning of each 'model' in the lyric.
Also, there should be no comma between brains and the size of models.
I hope this helps, let me know if you have any questions!