It is not necessarily wrong to start the Kārtik vow on full moon day. Haribhakti Vilāsa (16.183) says –
aśvine śukla-pakṣasya prārambhe hari-vāsare
athavā paurṇamāsītaḥ saṅkrāntau vā tulāgame
“(The Kārtik vow) starts on the Ekādaśī of the light quarter of the month of Āśvina. Or it can be started from full moon day, Saṅkrānti or from the day when the sun enters the Tulā-constellation.”
Sanātan Goswāmī’s commentary on the verse includes - hari-vāsare ekādaśyāṁ, tasya vaiṣṇava-pakṣatvād ādau nirdeśaḥ “Harivāsara means Ekādaśī – for Vaiṣṇavas that is the primary order”. So it is not wrong to start the Kārtik-vow on full moon, but starting on Ekādaśī is the first choice.