Texas titled vehicles by engine number until around 1954. It was common practice to change the engine number to match the vehicle. And of course this had to be done with factory replacement engines. We routinely bought short blocks and stamped the engine number on them to match the title. If that was how it was don back then, there is no reason to change the engine number. There is a form for correcting a title, but it can be a complicated procedure. Texas has a title only transfer where you only need a title and a bill of sale. What Texas is interested in is the $28 title transfer fee and the 6 1/4% sales tax. I was a licensed salvage dealer in Texas for many years, and Texas really doesn't care about vehicles that are not considered late model. All they want is the tax. When you bring an out of state vehicle into Texas, any state licensed inspector can verify the title against the vin. I am still a licensed inspector for commercial trucks. But if you do a title only, you don't need insurance or inspection. Once you get a title and if you want to put license on it, you need a safety inspection and insurance. For antique plates you don't need inspection or insurance, but you are very limited to how you can drive it.
Here is a list of forms available to get a Texas title. Most people make a more complicated thing than it really is.
http://www.txdmv.gov/forms
