I heard that Laxman is an avatar of Sheshnag? Is this true?
Yes, Shree Laxman and Balraam both are the avatars of Sheshnag. In Padma Puran there is a Chapter 1.5.11, The proof is mentioned here in this chapter. According this: When lord Mahadev was telling the story of Sri Ram to Parvati here he mentioned about Sri Laxman.
Lord Mahadeva told Parvati- "Dasharath requested Vashishth to perform the Jaatkarm Sanskar of Sri Ram. In course of time Kaikeyi gave birth to Bharat- an incarnation of Panchajanya conch. Similarly,
Sumitra gave birth to two sons--Laxman and Shatrughan. Laxman was the incarnation of Sheshnag while Shatrughan was the incarnation of Sudershan chakra.
Sri Ram and his brothers received education under the guidance of Sage Vashishth.
In a very short time they became proficient in all the scriptures and mastered various weaponaries. Although all the four brothers were very close to each other, yet Laxman could not bear to live in Ram's separation even for a moment and was specially close to him."
Link: Padma Purana
Another Proof, When Goswami Tulsidas was praying to Sri Laxman he also describes him as a Avatar of Sheshnag in below lines...
बंर्दौं लछिमन पद जलजाता। सीतल सुभग भगत सुखदात।।
रघुपति कीरति बिमल पताका। द्वंड समान भयउ जस जाका।।
सेष सहस्रसीस जगकारन। जो अवतरेउ भूमि भय टारन।।
Yes,this is true.
In the Treta Yuga, Lord Vishnu took seventh avatar as Lord Rama. Sheshnag accompanied him and took avatar as Lakshman–the younger brother.
Refer this good article.
Other answers provide Purāṇic angle, while this answer is focused singularly on Vālmīki Rāmāyaṇa.
tataḥ padmāpalāśākṣaḥ kr̥tvātmānaṁ caturvidham |
pitaraṁ rocayāmāsa tadā daśarathaṁ nr̥pam || 1.15.30 ||
Thereafter, the lotus petal-eye Lord was pleased to transform himself into four forms and to choose king Daśaratha as his father.
This śloka is from Vālmīki Rāmāyaṇa where Viṣṇu decided to divide himself fourfold ways at the pleading of devas and Brahmā. There is a great lot of discussion whether Viṣṇu made himself fourfold as Rāma, Lakṣmaṇa, Bharata an Śatrughna, or he himself incarnated along with his main ingredients like conch shell, disc, mace as the 4 brothers. This verse says that he 'agreed to make himself as fourfold.' Even then, there are discussions that Rāma is a 'full incarnation' (pūrṇa avatāra) as the Supreme is indivisible - on one account. And his brothers cannot be reckoned as his incarnation as they become parts of the whole but not the 'whole' itself he did not divide himself on the other account. As per tradition, it is believed that Viṣṇu didn't divide himself but incarnated in one piece, where his brothers are his paraphernalia.
The 'transforming into four forms' in aforementioned śloka (1.15.30) isn't present in the Sarga 15 of Bāla Kāṇḍa of critical edition of Vālmīki Rāmāyaṇa.
kausalyā śuśubhe tena putreṇāmitatejasā || 1.18.11 ||
yathā vareṇa devānāmditirvajrapāṇinā |
bharato nāma kaikeyyāṁ jajñe satyaparākramaḥ || 1.18.12||
atha lakṣmaṇśatrughnau sumitrā 'janayatsutau || 1.8.13 ||
vīrau sarvāstrkuśalau viṣṇorardhasamanvitau |
Kausalyā glowed with the undiminished lustre of her son, just as Āditi with her son Indra, the foremost among the gods and the wielder of thunder. As an incarnation of the fourth part of Viṣṇu, imbued with all virtues, and armed with the truth Bharata was born to Kaikeyī. Sumitrā gave birth to Lakṣmaṇa and Śatrughna who were heroic, skilled in the use of all weapons and endowed with the facets of Viṣṇu.
The Critical edition of Rāmāyaṇa on the other hand, has the similar portion in Sarga 17 of Bāla Kāṇḍa, instead of Sarga 18 of Bāla Kāṇḍa, and it supports the idea of fourfold incarnation of Viṣṇu more strongly. Its translation is as follows:
[6-12] Kausalyā gave birth to an illustrious son named Rāma, the delight of the Ikṣvākus. He bore the signs of divinity, for he was one-half of Viṣṇu. An immeasurably resplendent son, he glorified Kausalyā as does Indra, the foremost of gods and wielder of vajra. , his mother, Āditi. Kaikeyī bore a truly vailorous son named Bharata, one quarter of the incarnate Viṣṇu, endowed with every virtue. Sumitrā gave birth to two sons, Lakṣmaṇa and Śatrughna, heroes skilled in all weapons and infused with portions of Viṣṇu.
On the basis of this quote, Viṣṇu's incarnation were split in a ratio where Rāma was 1/2 Viṣnu incarnate, Bharata was 1/4 Viṣṇu incarnate and Lakṣmaṇa and Śatrughna had some fractions (<1/4) Viṣṇu incarnate.
Vālmīki Rāmāyaṇa (online links)
Vālmīki Rāmāyaṇa (Translation of BORI Critical Edition):
Goldman, R. P & Goldman, S.J.S. (Eds.) (2021). The Rāmāyaṇa of Vālmīki: The Complete English Translation. (R.P. Goldman, S.J.S. Goldman, R. Lefeber, S.I. Pollock, B.A. van Nooten, Trans.). Princeton University Press
There is another prospective to see.
राम नैवोद्धितो वीरो लक्ष्मणो विदधत्स्वकः । रूपत्रयं महद्वेषं लोकानां हितकाम्यया ॥ एकेन सरयूमध्ये प्रविवेश कृपानिधिः । सहस्रशीर्षा भगवान् शेषरूपी रसाश्रयः ॥ रामानुजश्चतुर्वाहुर्विष्णुस्सर्वगुहाशयः । ऐन्द्रं रथं समारुह्य वैकुण्ठमगमद्विभुः ॥ यानस्थो रघुनन्दनः परपुरीं प्रेम्णागमद् भ्रातृभिर्लोकानां शिरसि स्थितां मणिमयी नित्यैकलीलापदाम्। सौमित्रिश्च तदाकलेन प्रथमं रामाज्ञया वर्तते तेनैव क्रमकेन बन्धुमिलितो रामेण साकं गतः॥
With shri ram, laxman ji for sake of the world assumed 3 forms. One form sank in river saryu and became seshnag. Second form became became vishnu (that lies in the heart of everyone) and moved in chariot brought up by Indra. Third and most important form remained same and moved to saket lok with shri Ramchandra.
वैकुण्ठेशस्तु भरतः क्षीराब्धीशस्तु लक्ष्मणः । शत्रुघ्नस्तु स्वयं भूमा रामसेवार्थमागताः।।
(#इति_श्रीमद्_नारदपंचरात्रे narad pancharatra quoted in ramayan shiromani tika )
Bharat ji is lord of vaikunth, laxman is lord of ksheersagar and Shatrughana is bhouma purusha, all these vishnu(. Karnodakshyi vishnu,garbodashayi vishnu, ksheerodakshayi vishnu) have incarnated to serve param purusha swaym bhagwan shri Ramchandra.
पृथ्वि! स्थिरां भव, भुजंगम्! कुर्मराज!तदिदं द्वितयं दधीथाः। दिक्कुज्जराः! कुरुत तत्त्रितये दिधीषां रामः करोति हरकार्मुकमाततज्यम्।।
Laxman - Fearing that the earth and other worlds would sink down as soon as he offered the bow to Shri Ramchandra, he orders: - Hey Shesh! You keep holding this earth. Hey Kachhpraj! You keep handling both this earth and Sheshnag. O giants! You should be careful in wearing these three earth, Sheshnag and Kurma Raj because Shriram ji wants to offer the string of Shivaji's bow.
Again same thing is said in Ramcharitmanas
चौपाई दिसिकुंजरहु कमठ अहि कोला। धरहु धरनि धरि धीर न डोला॥ रामु चहहिं संकर धनु तोरा। होहु सजग सुनि आयसु मोरा॥ (#रामचरितमानस)
Laxman ji orders:- O giants! Hey turtle! Hey Shesh! Hey Varah! Be patient and hold the earth so that it does not move. Ram wants to break Shiva's bow. Everyone be careful after listening to my orders.
Note:- In hanuman natak and ramcharitmanas, laxmanji clearly orders sheshnaag, varah (avtar of Vishnu) and kacchap(kurma avtar of Vishnu). An avtar can't order avtari (one who takes incarnations here sheshnaag) and also order avtar of Vishnu. Surely, laxmanji is not avtar of sheshnaag
Again in vashisht samhita quoted in Manas piyush, clarifies that laxman ji is none other than vishnu himself