The OP asks: "Is/was animal sacrifice the only means of seeking forgiveness?"
Answer: No, animal sacrifice is not the only means in the Torah (Tanach) to seek forgiveness from G-d.
Examples of other things and ideas that can be used to seek forgiveness, from Scripture:
1) A bag of flour:
Leviticus 5:1-13 covers the sacrifice known as "Olah V"Yored" or a sliding scale sacrifice based on financial position of the supplicant. See verses 11 - 13.:
(11) And if he cannot afford two turtledoves or two young
pigeons, then he who sinned shall bring as his offering one tenth of an ephah of
fine flour for a sin offering; he shall not put oil on it, nor shall he place
frankincense upon it, for it is a sin offering. (12) He shall bring it to the priest, and
the priest shall scoop out a fistful as a memorial thereof, and burn it on the altar,
upon the fires of the L-rd; it is a sin offering. (13) The priest shall make atonement
for him from his sin which he had committed in any one of these [instances], and
he shall be forgiven; and [the rest of] it shall belong to the priest, like the meal
Flour does not contain animal parts or blood.
2) Incense :
Numbers: 17:12 "Aaron took as Moses had spoken and ran to the midst of the congregation, and behold, the plague had begun among the people. He placed the incense and provided atonement for the people."
Incense does not contain animal parts or blood.
3) Jewelry :
Numbers 31:50 "So we have brought an offering for G-d; what any man found of golden vessels, anklet and bracelet, ring, earing, and clasp, to atone for our souls before G-d."
4) Charity and loving kindness (as well as justice and humility):
Proverbs 16:6 "Through loving kindness and truth will iniquity be atoned; and
through the fear of the L-rd [you] depart from evil."
Daniel 4:24 [27 in Christian Bibles] "Only, O king, let my counsel be acceptable to
you, and your sins will be with charity removed, and your iniquities by showing
mercy to the poor; indeed, your tranquility will be prolonged."
Hosea 6:6 "For loving-kindness is what I desire, and not sacrifice; and knowledge
of G-d more than burnt offerings."
Micah 6:6-8 (6) "With what shall I come before the L-rd, bow myself before G-d on
high? Shall I come before Him with burnt offerings, with yearling calves? (7) Will
the L-rd be pleased with thousand of rams, or with ten thousands streams of oil?
Shall I give my firstborn for my transgression, the fruit of my body for the sin of my
soul? (8) Man has told you what is good; but what does the L-rd demand of you?
To do justice, and to love loving-kindness, and to walk humbly with your G-d."
Isaiah 1:11 and 17 - 18: (11) "Of what use to Me are your many sacrifices?" says the Lord; "I am sated from the burnt offerings of rams and the fat of well-fed cattle; and in the blood of bulls, and of sheep, and of male goats I do not delight." ......... (17) " Learn to do good, seek justice, help the oppressed; do justice to the orphan, plead [the case] for the widow. (18) Come now, and let us reason together," said the L-rd; "If your sins be as scarlet, they shall become as white as snow; if they be red as crimson, they shall become as wool."
5) Destroying and removing idols :
Isaiah 27:9 "Therefore, through this shall Jacob's iniquity be atoned for, and this shall be the fruit of his sin's removal: When he makes all the altar stones like ground chalk stones, and Asherah trees, and sun idols arise no more."
6) "Exile and punishment" :
Isaiah 40:1-2 (1)"Comfort, comfort my people, says your G-d. (2) Speak consolingly of Jerusalem and proclaim to her that her period (of exile) has been completed, that her iniquity has been forgiven; for she has received double for all her sins from the hand of G-d."
7) Repenting and changing your deeds :
Jonah 3:10 "And G-d saw their deeds, that they repented from their evil way; and G-d
relented of the evil, which He had spoken to do to them, and He did not do it."
No blood sacrifice appears in the Book of Jonah; just repentance, fasting, prayer, and better deeds.
Ezekiel 18:21-23 (21)"As for the wicked man, if he repents from all his sins that he committed.......(22) All his transgressions that he committed will not be remembered against him;..."
The OP asks:"...provide some verses which indicate that prayer and seeking forgiveness is also a viable means and that G-d will accept it."
Numbers 14:20 "And G-d said: "I have forgiven because of your words."
1 Kings 8:46-52 (46) "If they sin against You, for there is no man who does not sin,
and You will be angry with them, and deliver them to the enemy, and their captors
will carry them away captive to the land of the enemy, far or near; (47) And if they
take it to heart in the land where they were held captive, and repent, and make
supplication to You in the land of their captors, saying, "We have sinned, and have
done perversely, we have committed wickedness"; (48) And they return to You
with all their heart, and with all their soul, in the land of their enemies who led them
away captive, and pray to You toward their land, which You gave to their fathers,
[toward] the city which you have chosen, and [toward] the house which I have built
for Your Name; (49) Then You shall hear their prayer and their supplication in
heaven, Your dwelling place, and maintain their cause, (50) And forgive Your
people what they have sinned against You, and all their transgressions that they
have transgressed against You, and have mercy upon them before their captors,
so that they may have mercy on them; (51) For they are Your people, and Your
inheritance, whom You have brought out of Egypt, from inside the smelting
furnace of iron; (52) That Your eyes may be open to the supplication of Your
servant, and to the supplication of Your people Israel, to hear them whenever they
call to You."
See similar at 2 Chronicles 6:36-40.
2 Chronicles 7:13-14 (13) "If I shut up the heaven and there be no rain, and if I
command locusts to devour the land, or if I send pestilence upon My people. (14)
And My people, upon whom My name is called, humble themselves and pray and
seek My presence and repent of their evil ways, I shall hear from heaven and
forgive their sin and heal their land."
Hosea 14:2-3 (2) "Return, O Israel, to the L-rd your G-d; for you have stumbled in
your iniquity. (3) Take words with you and return to the L-rd; say to Him: "You
shall forgive all iniquity, and accept the good, and we will render [for] bullocks [the
offering of our] lips."
The Torah contains much more. I hope this helps. :)