The spelling of your name doesn't change in French. Shawnette Valery - would stay the same.
French has different pronunciation rules, but if a French speaker doesn't know English pronunciation rules, you would simply teach them the correct pronunciation.
The AW sound doesn't exist in French, so someone with poor or no English pronunciation skills would say SHAH instead.
Syllables in French get equal stress, so Valerie would be: vah (no short A sound as in valley either) -luh-ree (where UH is the schwa. In English, the schwa only occurs in unaccented syllables. In French, syllables usually get equal stress).
If the French speaker is pronouncing your last name only from seeing it written, it might come out as vahl-ree.
There is no way to write your two names according to French pronunciation rules that will produce the true pronunciations, since each name has at least one sound that French doesn't have.
Note also that the standard French R is not a true R at all. It's a different sound that English lacks. However, some dialects (and in many dialects that still have a French R, you will still hear a rhotic R depending on what sounds surround it) do use the rhotic R (which most dialects of English use, if the R is actually pronounced).
One of the regular users on here makes the false claim that French (in France) does not have true dialects. Even within France, there definitely are dialects and French is spoken in many countries around the world. They do not all speak Parisian French.
studied linguistics and phonology; native English speaker. taught French.