Oldsmobile straight eight engines were used in LaSalles for 1934, 1935, and 1936. For 1937, LaSalle went back to a Cadillac built V-8. The LaSalle V-8 was a 322 CID which was used until the end of LaSalle production in 1940. Cadillac cars used the same exact engine, except that the version used in Cadillacs displaced 346 CI, due to a larger cylinder bore. Very often, over the years, the LaSalle engines, when they were rebuilt, were bored out to the larger displacement. The larger and smaller engines are identical in appearance. Except for the pistons and rings, their parts are identical. The 346 lasted until the end of 1948 production.
I cannot help you with the casting numbers for 1937 engines. Cadillac cars in that era were identified by the chassis number, which can be found on the top of the frame, near the steering box. This number would not be found on the engine. In order to confirm the original engine number, you would need to have the build sheet for that car. Matching numbers are not an issue on prewar Cadillacs.
I have owned a 1940 LaSalle for many years. These are great cars. To my eye, they are better looking than their Cadillac brothers. I would look this car over like any other vintage car. They don't have any particular problems except that they are 85 years old. Look carefully. Remember that with every year that passes, improvements on these cars have gotten more expensive and harder to accomplish.
Good luck!