Sita does not get a headache in Valmiki, Kamban or Tulasidas. There are two places where Sita is shown as hugging Rama, repeat, Rama in Valmiki Ramayana. Occasion number one when Sita insists that she should also be taken to the forest, at the end of a three sarga long arguments, she gets exhausted and embraces Rama, drenching his chest with her tears.

इति सा शोक सम्तप्ता विलप्य करुणम् बहु |
चुक्रोश पतिम् आयस्ता भृशम् आलिन्ग्य सस्वरम् || २-३०-२२

22. saa = that Seetha; shokasantaptaa = burnt by grief; aayastaa = having exerted; vilapya = lamented; bahu = much; karuNam = in pity; aaliNgya = embraced; patim = the husband; chukrosha = cried; bhR^isham = a lot; sasvaram = with loud voice.

Seetha, burnt by grief, having got exerted, lamented much piteously, embraced her husband and cried a lot with loud voice.

(Ayodhya Kanda, Sarga 30, Sloka 22)

Occasion number two after Rama eliminates Khara, Dushana and their 14,000 strong army single handedly and returns to their hermitage, Sita runs to him and embraces him.

तम् दृष्ट्वा शत्रु हंतारम् महर्षीणाम् सुख आवहम् || ३-३०-३९
बभूव हृष्टा वैदेही भर्तारम् परिष्वजे |

39b, 40a. vaidehii = Seetha; shatru hantaaram = enemy, eliminator; maharSiiNaam sukha aavaham = for great-sages, rejoice, accorder; tam bhartaaram dR^iSTvaa = him, her husband, on seeing; hR^iSTaa babhuuva = became, rejoiced; pariSvaje = embraced.

And Seetha on seeing her husband, the eliminator of enemies and accorder of happiness to sages, embraced her husband Rama, rejoicingly. [3-30-39b, 40a]

The above are the ONLY two instances where embracing is mentioned in Valmiki Ramayana. (I have understood the question. I am coming to the point. Kindly read on.)

Kamban eliminates all scenes that involve a physical contact even between Rama and Sita. The above scenes are SKIPPED by Kamban.

Lakshmana's shyness for women is spoken of vividly by both Valmiki and Kamban. In Kishkindha Kanda, when the ornaments discarded by Sita were shown to Rama, Rama asks Lakshmana to see if they were the jewels of Sita. Lakshmana answers:

एवम् उक्तसः तु रामेण लक्ष्मणो वाक्यम् इदम् अब्रवीत् |
न अहम् जानामि केयूरे न अहम् जानामि कुण्डले || ४-६-२२
नूपुरे तु अभिजनामि नित्यम् पाद अभिवंदनात् |

22. evam uktaH tu raameNa = thus said, but, by Rama; lashmanaH = Lakshmana; vaakyam idam abraviit = sentence, this said; aham keyuure na jaanaami = I, bracelets, do not, know; aham na jaanaami kunDale = I, do not, know, earrings; nityam paada abhivandanaat = always, to feet, because I salute; nuupure tu = anklets, but; abhijanaami = I know them well.

When Rama thus said Lakshmana said this, "I know not the bracelets and I know not the earrings But because I always salute her feet I know these anklets well...[4-6-22]

When Lakshmana says that he has not seen any ornament of Sita, excepting her Nupur, how would he allow Sita to rest on his laps, even if Sita did not mind doing so! Mere absurdity, to put it very mildly.

Once again, this is also skipped by Kamban. He does not speak of this portion at all.

Then, when they were entering the forest in Aranya Kanda, in one place Rama tells Lakshmana to help Sita to get into the boat. Sita holds the hands of Lakshmana and gets into the boat. Helping to get into a boat while it is tugged at the banks necessarily involves Lakshmana having to hold her bodily, not just by the hands.

आरोह त्वम् नर व्याघ्र स्थिताम् नावम् इमाम् शनैः |
सीताम् च आरोपय अन्वक्षम् परिगृह्य मनस्विनीम् || २-५२-७५

75. nara vyaaghraH = "Oh; Lakshmana the tiger among men! Tvam = you; aaroha = get into; imaam = this; naavam = boat; sthityaam = stationed; (here); parigR^ihya = having helped; siitaam cha = Sita; mansviniim = the virtuous wife; aaropaya = step into it."

"Oh, Lakshmana the tiger among men! You get into the boat stationed here unhurriedly afterwards, having helped Sita the virtuous wife step into it."

स भ्रातुः शासनम् श्रुत्वा सर्वम् अप्रतिकूलयन् |
आरोप्य मैथिलीम् पूर्वम् आरुरोह आत्मवांस् ततः || २-५२-७६

76. shrutvaa = hearing; shaasanam = the command; sarvam = wholly; bhraatruH = of his (elder) brother; saH = that Lakshmana; aatmavaan = prudent; apratikuulayan = not resisting it; aaropaya maithiliim = having made Sita to ascend (the boat); puurvam = first; aaruroha = stepped into (the boat); tataH = afterwards.

Hearing the command completely of his elder brother, the prudent Lakshmana, by not counteracting it, made Sita to ascend the boat first and stepped into it afterwards.

This is one place (and only one) when Valmiki has explicitly stated that Lakshmana 'touched' Sita. Needless to say, this is absent in Kamban.

I have written on this as follows (in my character study)

In fact it was Lakshmana who helped her get into the boat, ordered by Rama. That goes without saying that Lakshmana would have had to hold her hand, help her to keep her balance by holding her tight et al. But that takes place as naturally as between a mother and son. Later, when Rama helps her to get into a ferry when crossing the Yamuna, the poet employs the epithet lajjamanam – shy, abashed – to describe her when Rama takes hold of her hand (in the presence of Lakshmana and others).

Having said all that, it is beyond imagination to say that Sita rested on the laps of Lakshmana because she had a headache. Even the very word headache does not occur in Valmiki only once, referring to Sita.

Even Rt Hon'ble Srinivasa Sastri who dealt with such places elaborately, does not mention this incident. It is not there in Ramayana. IT IS NOT THERE IN RAMAYANA.