$5k is about a dollar a day for 13 years; switching to natural gas is about 25% more efficient than fuel oil, co2-wise. Depending on how much heating you need that 25% could be vastly more impactful than a life without meat. Also, look into electric heating, depending on your electric supply that could be vastly more carbon efficient.
Also, around the house efficiency in the form of insulation and modern appliances is something that's usually super cost effective and carbon effective on a house from the 60s. It again will require up-front capital but is almost always a super smart move to do as soon as possible.
In some ways it's easier to be super self sacrificial and not eat meat, but it's important to look at the big picture. It really is about the numbers here; personal purity does nobody any good. Except on the political side; become a vocal single issue voter and never vote for a politician that doesn't have climate change part of their platform, and the same goes for political parties. Ultimately putting in mile carbon taxes will shift the market to do the right thing, but the politics make this impossible. Yes anti-economical too, negative externalities must be addressed by societal means, just like liabilities must be addressed through societal means.