Both are correct. Well, an -ώ verb either follows the old είς-conjugation or the more modern άς-conjugation, named after the ending of the conjugation of the 2:nd person singular.
μπορώ, μπορείς, μπορεί, μπορούμε, μπορείτε, μπορούν
περνώ/περνάω, περνάς, περνά/ περνάει, περνούμε/ περνάμε, περνάτε, περνούν/ περνάνε
The tendency is that more and more είς-verbs become άς-verbs, therefore the άς-verbs have reminiscences of the older conjugation as still living alternatives. I gave these old forms first because I respect age and they are still used though the modern alternative might be more popular (I use them myself)
«περνώ» is an old form as much as «περνάω». Now old form abolished for the first person plural is «περνώμεν». Instead of that either can be used «περνάμε» which is normal for verbs ending to «-άω» or, influenced by the verbs ending to «-έω», we can say «περνούμε», however it does not sound so good. BUT the opposite is horrible, that is to say «μποράμε» never for heavens sake. Os verbos ending to «-έω» are not influenced. So let's keep the «-άμε» for the verbs ending to «-άω».The phenomenon in greek is called συναίρεσις (syneresis) is old and it consists of the melding of two vocals to one as a part of one general inclination of the languages to avoid the hiatus (χασμώδια).
«διασχιζω» really means «to cross», as for «περνάω» there is the common expression «περνάω απέναντι» literally «pass across» usually used for a road, a river, a lake, in cases there is an obstacle and meaning «to go across» or simply to «cross». I suppose that, for the economy of speech, rests this «περνάω» alone without the «απέναντι». In any case «περνάω» is a «pass partout», take a look at this site https://el.glosbe.com/el/en/%CF%80%CE%B5%CF%81%CE%BD%CE%AC%CF%89