@Gerald Jordan
A few thoughts... assessed values are almost always low. Most taxing districts subtract a percentage off the actual value of the property to not have so many people complaining about their tax valuations. So anticipate your appraisal being higher than that. From your post we don't know what the new balance on the loan is (ie. was it a cash-out refi or just a rate and term refi). But if you said the house was worth $380,000 - just to pick a higher number - you would each be entitled to 'about' $190,000 if you were to sell the house (less expenses... which would probably be about 6-8% if you are talking 6% commission and closing costs. Using 6% for closing that would be $22,800 less you would net... so more like $357,200 - then less the balance of the mortgage - which if we used the $249k as the balance... that would be about $108,200 that you would normally split. The house would likely get a step-up basis in value I would think due to how it was transferred... so there would likely not be any capital gains taxes I wouldn't think.
It would be fantastic if you could assume the mortgage at the lower rate, because today's rate would be in the 6's and cost you a lot more to hold the property. That would be a phone call to the lender to determine. If you have other assets to distribute that you could compensate your sister with you might have a path to assume the mortgage yourself... but if you have to pay her off and the house holds the majority of the proceeds, it would seem like you might need to cash-out refi the house to be able to pay her off possibly? You would lose your good rate on the house at that point and likely be in the 7's on a rate if I had to guess.
Another path might be to rent it out and split the proceeds each month with your sister if that makes sense? The house would hopefully continue to appreciate, so would always be an asset for the two of you down the line.
Using real (ie. closer) numbers, you would just need to run the numbers and see if the property makes sense to hold given what rents are in your area, and what your expenses would be. For myself, for a rental, I am usually looking to make at least $300 a month after Principle, Interest, Taxes, Insurance, and a maintenance reserve of around $100. There are other expenses to think about... like Capex, and vacancy, and property management if you will be using that. But just run your numbers and see what they tell you. If it makes money in today's market, I would say try to keep it. If it doesn't... it would probably be best to sell it.
All the best, and condolences to your family!
Randy