What is the rule regarding accents on participles in the case of αντέχω? Why is the present active accusative accent on the antepenultimate ἀντίσχουσαν and the Aorist Active Participle a circumflex on the penultimate? ἀντισχοῦσαν?
αἱ δὲ παλαιαὶ διὰ τὴν λῃστείαν ἐπὶ πολὺ ἀντίσχουσαν ἀπὸ θαλάσσης μᾶλλον ᾠκίσθησαν…
If you switch ἀντίσχουσαν to ἀντισχοῦσαν, what would the meaning of the sentence be, if any?
The verb is (αντ-)ἴσχω, an alternative (collateral) form of (αντ-)ἔχω which exists only in the present paradigm. The feminine participle is ἴσχουσα, so accented, like ἔχουσα. (αντι-)σχοῦσα would be aorist participle. All perfectly normal. Compare e.g. λέγουσα (present) vs. εἱποῦσα (aorist).
The old cities built more away from the sea “on account of piracy holding out/persisting for a long time.” Aorist ἀντισχοῦσαν (so accented) would imply that piracy had held out for a long time but no longer did when the old cities were built. [I see that the medieval manuscripts have the aorist, whether -σχοῦσαν or -σχοῦσαι (the cities). But -ίσχουσαν is clearly right, describing the then current state of affairs.]