Karthika Masam is sacred days dedicated only to Lord Shiva.
Then Why Maha Shivaratri won't fall on Karthika Masam?
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Chapter 9 Vidyeswara Samhita of Shiva Mahapurana says
प्रसन्नः प्राह तौ नम्रौ सस्मितं भक्तिवर्धनः। ईश्वर उवाच
तुष्टोऽहमद्य वां वत्सौ पूजायाऽस्मिन्महादिने॥ ९॥
Lord Šiva, who always increases devotion, then said to Brahma and Visnu delightfully, "Let both of your remain happy.” Siva again said, "O Sons, this is the great day and I am happy with your worship for me.
दिनमेतत्ततः पुण्यं भविष्यति महत्तरम्।
शिवरात्रिरिति ख्याता तिथिरेषा मम प्रिया।। १० ।।
Since then, the day is considered to be most auspicious and yields merit. This is the date dear to lord Siva and is well known by the name of Sivarātri.
यत्पुन: स्तम्भरूपेण स्वाविरासमहं पुरा।
स कालो मार्गशीर्षे तु स्यादार्द्राऋक्षमर्भकौ।। १५ ।।
O boys, I had earlier appeared in the form of a pillar, it was the month of Margasirsa and the constellation was Ardra.
It is clear that Lord Shiva appeared as linga for the first time (in that Kalpa) during month of Margasira. So, Chaturdasi tithi during Krishna Paksha in month of Margasira is Maha Shivaratri.
Coming to Kartika Masa, Kartika Masa is dedicated to both Shiva and Vishnu. There's a Section called Kartikamasa Mahatmya (with 36 chapters) in Vaishnava Khanda of Skanda Purana describing importance of Vishnu worship in Kartika Masa, which you can read briefly in this answer.
Quoting "यत्पुन: स्तम्भरूपेण स्वाविरासमहं पुरा। स कालो मार्गशीर्षे तु स्यादार्द्राऋक्षमर्भकौ।। १५ ।। O boys, I had earlier appeared in the form of a pillar, it was the month of Margasirsa and the constellation was Ardra."
I think this refers to the Arudra Darisanam festival that Tamils celebrate on the Full Moon day in Ardra Nakshatiram in the month of Margazhi/Margarsheersham.
If this so, it is Shukla paksha Chaturdasi/Pournami not Krishna Chaturdasi