There are solid reasons to conclude that in DPs/NPs like they both, they must be the 'head' and both must be an adjunct. This stands whether you adopt the (in generative linguistics) now standard analysis of NP's as DP's (i.e., projections of a Determiner) initially proposed in S. Abney's (1987) thesis, or prefer a more traditional analysis of NP's as just NP's (one that can also be easily defended, even within P&PT).
Of course, if you adopt Abney's proposal, they both must be a DP, its head must be a D, and the only problem is that, under Abney's assumptions, both they and both can arguably be considered 'determiners'. Thus, are there reasons to choose one of them as head of the whole DP in your example?
If you opt for Abney's analysis, you must also adopt the rest of P&PT, and, in that case, the answer is 'yes, there are'. The most obvious one is that the P&PT of that period had already adopted the 'VP(/XP)-Internal Subject Hypothesis', according to which the subject of your sentence, They both, must initially be attached and receive its Theta Role as the highest argument of the VP fighting about corn in Spec VP. In other words, in PPT's 'underlying syntax' your sentence would start (with irrelevant structure omitted) as ...[ __ were [they both [fighting [about corn]]]. Since they is inserted from the lexicon with a nominative Case feature activated, it must eventually land in a specifier in which it can license/check... that feature, and, in your sentence, the only position in which it can do so is the specifier of Tense, i.e., at the overt level of representation, the position immediately preceding the finite verb were. Hence, they must 'move' upwards from Spec VP into the specifier of Tense, it cannot remain in its initial position under VP (as Economy would otherwise dictate!) or it would violate Case Theory, cf. * __ were they both fighting about corn; thus, your sentence will only be grammatical if at least they raises into Spec of Tense and gets its nominative Case feature licensed there, as in They were __both fighting for corn.
On the contrary, the determiner (or pre-determiner, or adverbial-quantifier, but let's leave that aside) both need not be assumed to have been inserted from the lexicon with either nominative or accusative Case, because, contrary to they, it can represent either (cf. I saw both [Accus.] vs. Both [Nom.] saw me). In such circumstances, a lexical item may be inserted with an unvalued Case-feature, and, as a consequence, need not occupy a structural position in which any particular Case is licensed. Consequently, contrary to they, both need not independently move anywhere to satisfy Case Theory and can stay in situ, as in They were __ both fighting about corn. What's more, since both need not check Case, granted Economy, it shouldn't be able to move on its own anywhere, and, indeed, if it does move into the available Spec of Tense, the result (i.e., * Both were they__ fighting about corn) becomes ungrammatical - on two scores, a) because the raising of both would be gratuitous (and violate Economy, in current terms), and b) because it would also prevent the necessary raising of they into Spec Tense in search of a legitimate nominative Case value, which will violate Case Theory.
As a consequence, either both stays behind in situ (as a so-called 'floating quantifier') and the result is They were __ both fighting about corn, or it accompanies they when it raises into Spec Tense and the result is They both were __ fighting about corn. If it does, They both must have been a unitary constituent at underlying structure, or Move would not have been allowed to raise them together. If it does not, and only They moves into Spec Tense, then they both cannot have been a unitary DP at underlying structure, because, if it had, they (a D/DP) would not have been able to raise on its own out of a higher DP without violating the A-over-A Condition (aka 'minimality', aka 'shortest movement' and, ultimately, Economy).
It follows that they both must be analysable as a unitary DP or as two separate constituents DP + QP? depending on whether they both raises as a unit or they raises on its own leaving both behind. That is why not all linguists accept Sportiche's Q-Float analysis, but we can ignore that detail now and concentrate on the former case, because, if they both is not a constituent, your problem (which of them is the head?) automatically dissolves.
When they both is a 'deep' unitary DP and lands in Spec Tense, then, there remains the question whether the head of that DP is they or both, and why, i.e., what primarily motivated your question.
Now, suppose the head is both. In that case, given the surface order, they must be a specifier, but then its Case feature will remain unlicensed, because although Tense licenses the nominative Case feature of its DP specifier, if the head is both, that licensed Case feature will 'percolate' down into it, but not into its specifier. On the contrary, if the head of they both is they, the nominative Case feature that Tense assigns to the whole DP will 'percolate' (cf. the Head Feature Convention of GPSG/HPSG) into its head, the Case feature of they will be licensed, and the sentence will be grammatical. The fact that it is, then, forces us to conclude that the head of they both cannot but be they.Q.E.D.
If,instead of Abney's theory of the NP as a DP, you prefer the traditional NP analysis, a very similar reasoning leads to the same conclusion: if they is the head of they both, the Case feature assigned/licensed on the whole NP in Spec Tense will 'percolate' down into its head, they will satisfy Case Theory and the sentence will be grammatical. If, on the contrary, they is not the head, but a specifier, it will violate Case Theory and the sentence will be out.
Finally, to the extent you should entertain alternative analyses under which both is an adverbial quantifier attached to the VP but does not form a unitary constituent with the subject they, both they and both are heads of their respective phrases (DP, QP) and your problem disappears.