You need to ask your county assessor in Texas. Every state has different rules.
(In New York, I believe the seller is responsible for up to 3 years of property taxes when property is sold to a non-exempt person or organization, although it is usually part of the contract that the buyer reimburses the seller for these taxes at closing. Property owned by an EO--exempt organization--is only exempt from property taxes if it is being by the EO for an exempt purpose.)
For income tax purposes, an exempt organization is still subject to Unrelated Business Income Tax on income not related to its charitable purpose. For example, if a church owns a parsonage, but the pastor has their own house so the church rents it to ordinary renters as a way to make money, the house may not be exempt from property tax and the income may be taxable unrelated business income.
Either EO in your question might be subject to capital gains income tax if the land sold for more than they paid for it, depending on whether it is deemed as unrelated business income.
If you are just curious, ask your county assessor what happens when an EO sells property. If you are a member of the EO responsible for their finances, you may want to get hold of an accountant before you end up in an audit over UBTI.
