If the Old testament forbids the eating of flesh and blood, does it follow that Jesus is advocating something against God’s law when he commands us to eat His flesh and blood in the new testament? And if non-Catholics preach that eating blood is disobedience to God’s precept, then why are they patronizing McDonald’s fried chicken and cafeteria’s beefsteak?

Some of the non-Catholics allude to Genesis 9:4 ("Only flesh with its lifeblood still in it you shall not eat") and Acts 15:28-29 ('It is the decision of the holy Spirit and of us not to place on you any burden beyond these necessities, namely, to abstain from meat sacrificed to idols, from blood, from meats of strangled animals, and from unlawful marriage. If you keep free of these, you will be doing what is right.') as references that according to them support the depravity of eating blood.

Is this claim factual?

Yes and no.   Yes, it’s true that it is indeed proscribed in the Old Testament but the prohibition against eating blood found in the Old Testament was a discipline associated with the covenant between God and Noah, and incorporated into the Mosaic Covenant.  No, because when Jesus established the New Covenant through His death and resurrection, the disciplines of the Old Covenant became unnecessary. The discipline was preserved for a brief time in the early Church to protect new converts from scandal, but was not linked with the theological understanding of grace. When taken in context, it can be found that the consumption of blood is not outlawed and, moreover, it is necessary in the Eucharist.

Reconciling these Scriptural feferences that are seemingly contradictory

FIRST, any divine command that comes later changes divine command that came earlier.  When our Lord declared all foods clean in Mark 7:18-19 (Jesus said to the disciples, "Are even you likewise without understanding? Do you not realize that everything that goes into a person from outside cannot defile since it enters not the heart but the stomach and passes out into the latrine?" Thus He declared all foods clean. If God opposes the eating of blood as part of human food and thus naturally evil, why did Jesus reply to eat His flesh symbolically and to drink His blood symbolically? Otherwise, if this is so, He would be commanding us to act out symbolically a naturally evil deed as part of a sacred worship service. 

SECOND, the command against eating blood, like all of the Old Testament dietary regulations, has passed away for, "These are shadows of things to come; the reality belongs to Christ.  Let no one, then, pass judgment on you in matters of food and drink or with regard to a festival or new moon or sabbath (Colossians 2:17,16) 

The mention of not eating blood in Acts 15:20 ("Tell them by letter to avoid pollution from idols, unlawful marriage, the meat of strangled animals, and blood) was a pastoral provision suggested by James to keep the Jews from being scandalized by the conduct of Gentile Christians.  Evidently, these pastoral provisions were only temporary.  Jews were predisposed to abstaining from idol meat, but later Paul says idol meat is alright as long as it doesn't scandalize others (Romans 14:1-14;  1 Corinthians 8:1-13. 

Eating Blood in General

In Leviticus 17:10-13, anyone whether of the house of Israel or of the aliens residing among them, who partakes of any blood, would be cut him off from among Moses' people, explaining that life of a living body is in its blood, and so in Deuteronomy 12:23-24. 

However, St. James explained why it is so: For Moses, for generations now, has had those who proclaim him in every town, as he has been read in the synagogues every sabbath"(Deuteronomy 12:23-24).  It would thus help Jewish Christians if Gentile Chrsitians stopped from particular things thought to oppose the Mosaic law.

But why is the Precious Blood almost never distributed at Mass if Christ Himself orders the reception of both species?

By not distributing the Precious Blood, the priest would seem to be ignoring Jesus' command initiated the Eucharist at the Last Supper: "Take and eat: this is my Body. Then He took the cup, gave thanks, and gave it to them saying, Drink from it all of you, for this is my Blood of the covenant, which will be shed on behalf of many for the forgiveness of sins."

From this command, Jesus did this so as to perpetuate the sacrifice of the Cross throughout the ages until He should come again, and so entrust to His beloved Spouse, the Church, a memorial of His death and resurrection: a sacrament of love, a sign of unity, a bond of charity, a paschal banquet in which Christ is consumed, the mind is filled with grace, and a pledge of future glory is given to us" (Catechism of the Catholic Church CCC 1323).

In line with this, an individual receives the fullness of grace of the sacrament whether just receiving the Sacred Host alone, or the Precious Blood alone, or both together (Catechism of the Catholic Church CCC 1390). Since a person receives "the whole and entire" Christ under each species, the Church is obedient to the command of the Lord to eat His Body and drink His Blood by just offering one species to the congregation, even though this partaking is best indicated when both species are offered and consumed. For this reason, the Church has not required that both species should always be offered.

In the very early Church, Holy Communion under both species was distributed. The practice, however, gradually changed to avoid "some dangers and scandals," from the spillage of the Precious Blood in its distribution, to health concerns from sharing the same cup, to intoxication, and to absconding with the sacred vessels.

Even Non-Catholics who preach that eating blood is immoral, also eat blood!

How?  It is not possible to avoid eating blood.  Although the meat of a pork/beef/lamb, etc., is drained after it is slaughtered, some of it which can be found from in between of the tissues.  That is why it remains reddish as you chop the meat because of its blood content.  Hence, non-Catholics also eat blood obviously because of the fried chicken they eat from McDonalds, Jollibee, Kenny Rogers and KFC; and the pork steak they enjoy in their favorite cafeteria - these meats contain blood before they are cooked.

CONCLUSION:

Therefore eating blood does not oppose God's precept,as Jesus says, everything that goes into a person from outside cannot defile since it enters not the heart but the stomach and passes out into the latrine (Mark 7:18-19)