Well refuting or answering questions of anti-Islam or anti-sunnah etc. websites is not the purpose of this site. I'm answering this with a focus on source identification or reference-request and check of the authenticity of the source with a small insight into hadith interpretation.
An anti sunnah-website that misuses Sunnah to show its contradiction
This hadith (is hadith the right term here?) can be found in ibn Sa'ad's Tabaqat and in Sunan ad-Darimi and in Mosannaf 'Abdurrazaq.
Note that none of these sources have reported the bold part of this sentence:
Once Al-Abbas entered the room, he put his hands at once at his nose, and said to the gathered men outside
Nor did al-'Abbas say:
His body began decomposition
but he said:
وَإِنَّهُ لَبَشَرٌ وَإِنَّهُ يَأْسُنُ كَمَا يَأْسُنُ الْبَشَرُ
He is nothing but a human being that (will) decompose(s) like any other human being.
So these are clear additions and fabrications of the author of this article. And this is not the only deviation from the text as reported by the above sources! there's no mentioning of the a-Taqeefah council all this is just a "nice addition" to create a picture that might be close to truth, but includes some imagination and description of a situation that might lead to false conclusions like if the rest of the story or the main goal (a falsified statement) will be accepted as a truthful explanation and hold as an argument and evidence to show the contradiction with another statement that most people rely on and refute it. Of course not referencing a source is another tool to keep this false statement upright, as they may say no the source is not such nor such.
Authenticity based on the given sources
As for ad-Darimi (see here in Arabic) and according to islamqa fatwa #282133 (so far in Arabic only) ibn Sa'ad they reported it via:
Sulaiman ibn Harb سليمان بن حرب (trustworthy) -> Hammad ibn Zayd حماد بن زيد (trustworthy)-> Ayyub as-Sakhytani أيوب السختياني (trustworthy) -> 'Ikrimah عكرمة (trustworthy)
While 'Abdurazzaq only deviated slightly to this narrator chain by reporting it via:
Ma'amar (for further details on this narrator you may refer to What is the authenticity of the following narrative in which Aisha RA had a girl remove the bells from her anklets?) -> Ayyub as_Sakhtyani -> 'Ikrimah.
The issue here is that 'Ikrimah is a tabi'iy (who hopefully is the companion of ibn 'Abbas as else there's a second 'Ikrimah among the followers -tabi'iyn- who met one single sahabi -according to imam a-Dhahabi who's name is al-Hirmas ibn Ziyad الهرماس بن زياد who died 90 a.H.) and he never met Al-'Abbas himself and we don't know who was the intermediate that told him this encounter at least according to these sources.
There's however a report of at-Tabarni that fixes the issue in this narrator chain: as it states that 'Ikrimah has reported from ibn 'Abbas according to this fatwa on islamweb #143643 this was reported via:
Ibn 'Oyyanah سفيان بن عيينة -> Ayyub as-Sakhtyani -> 'Ikrimah -> Ibn 'Abbas -> Al-'Abbas
This chain would basically (assuming it is connected as it seems to be the case -I didn't find it yet-) reach the condition of al-Bukhari.
One issue I found in the content of this narration is that it suggests that Muhammad () was buried Wednesday night while sahih narrations report that he was buried Tuesday see for example in al-Muwatta' and Sahih al-Bukhari but ibn Majah reported that this happend on Wednesday (see here).
Explanation and resulotion of the pretended contradiction
As for the statement of al-'Abbas:
"Bury your friend Muhammad (fast), for he's (nothing but) a human being (of) that (his body) will decompose like any of the human beings"
Most scholars like ibn Hajar explain it as an ijtihad of al-'Abbas not a as something he experienced or witnessed at this moment. This is only a conclusion al-'Abbas made without having prior knowledge, so he thought that what happens to the body of any human being will happen to the body of the prophet(). So there's no such thing like a contradiction to the hadith which was compiled by the authors of the Sunan and imam Ahmad in his Musnad as these ahadith report a statement of the prophet () himself -who should know better- while this report just displays a conclusion of al-'Abbas.
Also be aware that the statement or better speech of al-'Abbas even show this as it goes much further he made his call or alert to bury the body of the prophet () fearing that decomposition would happen to this body of a person of high esteem not because there already were any signs of decomposition like an odor as the additions may suggest.
Here's the hadith as stated in a-Darimi's Sunan:
As I'm not a professional translator nor a native English speaker.
I'm translating only some parts -for example italic parts of the original text are left untranslated- of this hadith as these translations are of my own take them carefully!
توفي رسول الله صلى الله عليه وسلم يوم الاثنين فحبس بقية يومه وليلته والغد حتى دفن ليلة الأربعاء
The Messenger of Allah Allah's prayers and blessing be upon him died on a Monday and was kept (like he was) for the rest of the day and night and the next day (too) until he was buried on Wednesday night.
وقالوا إن رسول الله صلى الله عليه وسلم لم يمت ولكن عرج بروحه كما عرج بروح موسى فقام عمر فقال إن رسول الله صلى الله عليه وسلم لم يمت ولكن عرج بروحه كما عرج بروح موسى والله لا يموت رسول الله صلى الله عليه وسلم حتى يقطع أيدي أقوام وألسنتهم فلم يزل عمر يتكلم حتى أزبد شدقاه مما يوعد ويقول
Some said that the Messenger of Allah Allah's prayers and blessings be upon him did not die, but his soul was raised as it was the case with the soul of Musa. 'Omar stood up and said (to the gathered people): the Messenger of Allah Allah's prayers and blessings be upon him did not die, but his soul was raised as it was the case with the soul of Musa, by Allah the Messenger of Allah Allah's prayers and blessings be upon him will not die until he will cut off the hands of some people and their tongues ...
فقام العباس فقال إن رسول الله صلى الله عليه وسلم قد مات وإنه لبشر وإنه يأسن كما يأسن البشر أي قوم فادفنوا صاحبكم
Al-'Abbas stood up and said (to the gathered people): indeed the Messenger of Allah Allah's prayers and blessings be upon him died and he is nothing, but a human and will decompose like any human, oh people bury your companion.
فإنه أكرم على الله من أن يميته إماتتين أيميت أحدكم إماتة ويميته إماتتين وهو أكرم على الله من ذلك أي قوم فادفنوا صاحبكم
As he is too noble by Allah to be caused to death twice, does HE cause one of you to death once and cause him to death twice while he is too noble by Allah to be treated like this oh people bury your companion!
فإن يك كما تقولون فليس بعزيز على الله أن يبحث عنه التراب إن رسول الله صلى الله عليه وسلم والله ما مات حتى ترك السبيل نهجا واضحا فأحل الحلال وحرم الحرام ونكح وطلق وحارب وسالم ما كان راعي غنم يتبع بها صاحبها رءوس الجبال يخبط عليها العضاه بمخبطه ويمدر حوضها بيده بأنصب ولا أدأب من رسول الله صلى الله عليه وسلم كان فيكم أي قوم فادفنوا صاحبكم قال وجعلت أم أيمن تبكي فقيل لها يا أم أيمن تبكي على رسول الله صلى الله عليه وسلم قالت إني والله ما أبكي على رسول الله صلى الله عليه وسلم ألا أكون أعلم أنه قد ذهب إلى ما هو خير له من الدنيا ولكني أبكي على خبر السماء انقطع قال حماد خنقت العبرة أيوب حين بلغ ها هنا