Kārtik - from Ekādaśī or full moon?

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.

Scriptural knowledge dormant in the heart?

Bhaktivinode writes in chapter 3 of his Vaisnava-siddhanta-mala):

Q. How can one know the truth about the Lord (bhagavat-tattva)?

A. He can be known by the soul’s own self-manifest innate knowledge (svataḥ-siddha-jñāna).

Q. What is self-manifest knowledge (svataḥ-siddha-jñāna)?

A. There are two types of knowledge: (1) self-manifest, or svataḥ-siddha-jñāna and (2) that which depends on the senses, or indriya-paratantra-jñāna. Self-manifest knowledge is that which is inherently a feature of the pure soul’s original spiritual form, or svarūpa. It is eternal, just like all spiritual reality. This inherent self-manifest knowledge is called Veda or āmnāya. This pure knowledge, or svataḥ-siddha-jñāna, has appeared along with the conditioned soul in the material world in the form of the Vedas, namely the Ṛg-veda, Yajur-veda, Sāma-veda, and Atharva-veda. Ordinary people collect knowledge of various objects by the help of their material senses. This is called indriya-paratantra-jñāna.

Q. If Bhagavān is attained through the soul’s own self-manifested knowledge (svataḥ-siddha-jñāna), then what is the necessity of studying the Vedic scriptures?

A. It is true that the Veda is present in every pure spirit soul in the form of svataḥ-siddha-jñāna. However, that self-manifest Vedic knowledge awakens in the heart in proportion to the soul’s freedom from bondage. It is manifest completely in some, and present in a covered form within others. In order to make that svataḥ-siddha-jñāna available to everyone, the Vedas have appeared in this world."

If he means to say with this that śāstrik knowledge is dormant in the heart of the conditioned soul then I'm afraid that is not correct.

Śrīmad Bhāgavata [11.22.10] says -

anādy-avidyā-yuktasya

puruṣasyātma-vedanam |

svato na sambhavād anyas

tattvajño jñānado bhavet

"Because a person whether covered or not covered by beginningless ignorance cannot realize ātmā on his own, there must a separate Supreme Lord who knows and gives knowledge."

Śrīla Śrīdhara Swāmi comments on this verse -

svato na sambhavati, anyatas tu sambhavāt. svataḥ sarvajñaḥ parameśvaro’nyo bhaved 

“It is not possible [to have or attain knowledge] on one's own - it is only possible from others and that ' other' is the Supreme Lord.”

[Quoted by Śrīla Jīva Goswāmi in Paramātma Sandarbha 44]