You can find the answer on vosdroits.service-public.fr (in French). I summarize and translate the relevant points (omitting cases that don't apply to you).
Le locataire qui donne congé à son bailleur reste tenu au paiement de son loyer pendant les 3 mois qui suivent la réception de son courrier par le propriétaire.
If the tenant notifies the landlord of his decision to end the rental contract, 3 months of rent are due starting on the date when the landlord receives the notification mail.
(1 January being a public holiday, if you sent your letter on 1 January, it was only sorted on 2 January, so a delivery the next day on 3 January is normal, even fast.)
Le délai court à compter de la date de réception effective de la lettre de congé. Ainsi, par exemple, une lettre de congé reçue le 5 septembre fait courir un préavis jusqu'au 5 décembre à minuit s'il est de 3 mois (et non jusqu'au 31 décembre).
The 3 month delay runs from the date when the notification letter was effectively received. For example, if the notification letter was received on 5 September, then the delay runs until 5 December at midnight.
lors du dernier mois du délai de préavis, le loyer dû par le locataire est déterminé proportionnellement au nombre de jours restant à courir dans le mois.
Exemple : la lettre de congé est reçue par le propriétaire le 5 juin. Le locataire est redevable du loyer jusqu'au 5 septembre, et paiera pour le mois de septembre les 5/30ème du loyer et des charges de ce mois.
For the last month, the rent due is calculated proportionally to the number of days remaining in the month. Translating the example to your date: since the letter was received on 3 January, you owe the rent until 3 April, so you owe 3/30 of a month's rent for April.
That page doesn't discuss the case when the état des lieux is not made until after the rental contract expires. I can't find an authoritative source for this, but here is a non-authoritative source which I believe is correct. Normally, the rental contract cannot end until you give back the keys to the landlord. The état des lieux can be established at this date, or later (provided the dwelling remains empty); even if it is established later, you do not owe rent for the period after you gave back the keys if the 3 month delay is over. If you keep the keys after the end of the rental period, then you do keep owing rent, unless you can prove that you vacated the premises and that the landlord is sole responsible for the key handover having been delayed.
Thus, if you gave back the keys on 3 April, you do not owe rent past this date, even if the état des lieux was delayed until 6 April because the landlord was not available earlier. If you stayed on the premises, you do owe rent. If you vacated the premises on or before 3 April but did not give back the keys until 6 April, then you only owe rent until 3 April, but you'll need to prove that you vacated the premises and that the it is the landlord's fault for not giving you an opportunity to give the keys back until 6 April.