It depends. If the shipping cost is relatively low (less than $10), I would recommend including that in the product price and offer free shipping to take advantage of the search ranking boost. Offering free shipping can also increase conversion rate (unexpected costs like shipping is one of the major reasons for abandon cart).
However, for higher shipping costs like over $10, then it’s understandable to include it as a separate cost, or put part of the shipping cost into the product price.
I would recommend doing this. Multiple studies have found that prices that end with a “9” or other odd numbers increase sales.
This book - Priceless - shows that there is a 24% increase in sales for prices ending in odd numbers compared to nearest rounded numbers
An experiment done in an MIT paper shows that out of three prices for the same women’s clothing item - $34, $39 and $44. The item that’s priced $39 sold the most.