There is no basis for the belief that it is not permissible for a woman to go near the birds during her period. There is no connection between the two matters. What you must do is give up such beliefs. There are certain rulings which apply to the menstruating woman but such matters are not among them and are not at all connected to them.
The scholars are unanimously agreed that it is permissible for a woman to slaughter animals (for food) when she is menstruating. See al-Mughni, 13/311.
Everyone who keeps these birds or any other creatures is obliged to take good care of them. If he is not able to do that then he should not keep them, and he should sell them or slaughter them if they are the kinds of animals or birds that can be eaten. There is a stern warning issued to the one who keeps an animal and does not feed it or give it water until it dies.
Ibn Qudaamah al-Maqdisi (may Allaah have mercy on him) said:
Whoever owns an animal must take care of it and spend on whatever it needs of food or appoint someone else to look after it, because of the report narrated by Ibn ‘Umar, who said that the Prophet (peace and blessings of Allaah be upon him) said: “A woman was punished because of a cat which she detained until it died of starvation, because she did not feed it or let it go and eat of the vermin of the earth.” Agreed upon. If he refuses to spend on it, he should be forced to do so; if he refuses or is unable to do that, he should be forced to sell it or to slaughter it if it is an animal that may be slaughtered (for food). End quote.
Al-Shawkaani said, commenting on the hadeeth about the woman who detained the cat:
It is understood from this hadeeth that it is haraam to detain cats and similar animals without giving them food and water, because this is tormenting the creation of Allaah, which has been forbidden by the Lawgiver… al-Nawawi said: It seems that she (this woman) was a Muslim, but she entered the Fire because of this sin. End quote.
Nayl al-Awtaar, 7/7
It says in al-Mawsoo’ah al-Fiqhiyyah (5/119, 120):
It is permissible to keep animals for beneficial purposes, such as guarding (property etc), listening to their voices and enjoying looking at them, but the one who keeps them is obliged to give them food and drink, because of the sanctity of life. End quote.
In the answer to question no. 48008 it says:
It is permissible to keep birds and similar creatures in cages for pleasure, especially if that is because of their appearance or their voices, subject to the condition that one feeds them and gives them water. End quote.
See question no. 48008 for more detailed information.
If your excuse for not feeding these birds until they died is that you forgot and did not do it deliberately, then this is an excuse which is acceptable according to sharee’ah. We ask Allaah to forgive you for that. Allaah says (interpretation of the meaning):
“Allaah burdens not a person beyond his scope. He gets reward for that (good) which he has earned, and he is punished for that (evil) which he has earned. ‘Our Lord! Punish us not if we forget or fall into error…’”
And Allaah has answered this prayer of the believers and says, “I have done that,” as was narrated by Muslim (126) from the hadeeth of Ibn ‘Abbaas (may Allaah be pleased with him).
And Allaah says (interpretation of the meaning):
“And there is no sin on you concerning that in which you made a mistake, except in regard to what your hearts deliberately intend”
Since you did not deliberately intend to kill them, there is no sin on you, in sha Allaah.
And Allaah knows best.