I'm confused about which Asr time I should follow. Can I pray in both of these times. If not then which has more evidence to support it, that I should follow it?
Differences in the Mathab is a mercy for us according to the following Hadith:
اختلاف أمتي رحمة
The only issue is that the above hadith is weak, but I personally think it true since differences give us an exit when we need it. Take the following authentic hadith for example:
أنّ رجلاً أصابه جرح في عهد رسول الله (صلى الله عليه وسلم) فأُمِر بالاغتسال، فاغتسل فمات، فبلغ ذلك رسول الله (صلى الله عليه وسلّم) فقال: قتلوه قتلهم الله، ألم يكن شفاء العيّ السؤال.
A person with a head injury during the time of the Prophet PBUH was asked to wash his head for Wudu, the prophet PBUH later said to the people who told him so "They killed him, Allah may kill them", wasn't it a cure to just ask.
My personal take from the Hadith is that nothing is set in stone and if there is a way something can happen to make things easier for you, you should do as long as you have the correct interruption.
Personally, I think if you need to pray Asr at a certain time and it is convenient for you, go ahead. Allah meant this deen to be easy and not complicated. Sometimes I pray Friday in one Masjid 1 hour before Zhur and sometimes in another Masjid 2 hours after. It is all on what is best to you.
Well if you want to pray and don't have any headache take the Hanafi rule. More detailled answer is here: http://www.isgr.org/asrpray.htm
This is also supported by a Hadith in Muwata' Malik
Yahya related to me from Malik from Yazid ibn Ziyad that Abdullah ibn Rafi, the mawla of Umm Salama, the wife of the Prophet, may Allah bless him and grant him peace, asked Abu Hurayra about the time of the prayer. Abu Hurayra said, "Let me tell you. Pray dhuhr when the length of your shadow matches your height, asr when your shadow is twice your height, maghrib when the sun has set, isha in the first third of the night, and subh in the very first light of dawn," i.e. when the dawn has definitely come.
وَحَدَّثَنِي عَنْ مَالِكٍ، عَنْ يَزِيدَ بْنِ زِيَادٍ، عَنْ عَبْدِ اللَّهِ بْنِ رَافِعٍ، مَوْلَى أُمِّ سَلَمَةَ زَوْجِ النَّبِيِّ صلى الله عليه وسلم أَنَّهُ سَأَلَ أَبَا هُرَيْرَةَ عَنْ وَقْتِ الصَّلاَةِ فَقَالَ أَبُو هُرَيْرَةَ أَنَا أُخْبِرُكَ صَلِّ الظُّهْرَ إِذَا كَانَ ظِلُّكَ مِثْلَكَ وَالْعَصْرَ إِذَا كَانَ ظِلُّكَ مِثْلَيْكَ وَالْمَغْرِبَ إِذَا غَرَبَتِ الشَّمْسُ وَالْعِشَاءَ مَا بَيْنَكَ وَبَيْنَ ثُلُثِ اللَّيْلِ وَصَلِّ الصُّبْحَ بِغَبَشٍ . يَعْنِي الْغَلَسَ .
But consider that the differences could sometimes come from mistakes in calculation or extreme carefulness. For example in the internet you can find many different calculation methods and many prayer times for the same prayer sometimes with diffrences of an hour or more!
Therefor one should take the one of his/her trust!