There's a verse in Srimad Bhagavatam that explains this directly.
kāma-lobhādayaś ca ye
ceta etair anāviddhaṁ
sthitaṁ sattve prasīdati
Translation: As soon as irrevocable loving service is established in
the heart, the effects of nature's modes of passion and ignorance,
such as lust, desire and hankering, disappear from the heart. ...
I. Can one with predominance of Rajas and Tamas worship Lord Vishnu?
Yes. Those with Rajasic and Tamasic Guna predominance can become devotees of Vishnu. Where Vishnu is considered the Supreme, though in his Saguna form as Vishnu he is sattvic, yet as the Supreme God he is the Lord of Maya and above these gunas which are in fact a product of Maya. ŚB 6.1. ...
Rama without a doubt followed both Rajdharama and Patidharma.
Just the day prior to Sita's exile Rama asked what he can do for her to which Sita replied the following-
Beholding his consort glowing with beauty, Raghava experienced an unequalled delight and exclaimed “It is well!” then he addressed the lovely Sita, who resembled a daughter of the Gods, and ...