About the usage of 'embarrassed': with a preposition, and/or without?

Which sentence is grammatically correct? Or are both? Neither?

• I felt embarrassed how little I knew about my own country.

or

• I felt embarrassed by how little I knew about my own country.

Though both are in use, the variant with by is far more frequent when it is an exclamative with how little following embarrassed. The other popular choices of preposition here are at and about.

Even though embarrassed how little doesn't show up on the ngram, the examples below don't seem entirely off to me, though certainly a little informal.

That’s half the battle. And if you can, remember to read some grown-up books for yourself, too – I’m embarrassed how little time I make for reading now. (The Sun)

I had clients coming back from overseas and I realized how little I knew about what they had gone through and the more I talked to them…I was embarrassed how little I knew. (Shepherd Express)

With subordinate exclamatives there is often a range of prepositions seen: with, over, as to can also be found in similar contexts.

So the Tigers were left hoping that the Big 12 would even stay in existence, which it did, but they have to be embarrassed with how little regarded they are on the national scene. (National Football Post)

When I was Showhat's age I would never have come to a restaurant like this, I'd have felt too intimidated and embarrassed over how little I knew about being Eritrean. (Interview in The Guardian)

“Frankly, back then, I was somewhat embarrassed as to how little I knew about Michigan’s tribal courts, and the fact that the vast majority of state court judges were equally uninformed about it,” he remarked. (Quote from retired justice Michael F. Cavanagh)

Incidentally, the use of embarrassed (OED: Feeling, expressing, or characterized by uncertainty; perplexed, confused, bewildered. Now rare.) followed directly by an interrogative, often a to-infinitival, fell out of fashion over a century ago.

In fact, they addressed themselves to him, and demanded his orders. Ferdinand, greatly embarrassed how he should reply to this message, declared that this question should be treated and resolved upon at Bayonne, together with those various points which were about to occupy the two sovereigns of France and of Spain. (History of the Consulate and the Empire of France Under Napoleon, Volume 2, Adolphe Thiers, 1893)

Jessie was rather embarrassed how to reply, but, at length, she said she thought it would be right to tell them of all they did wrong, and to persuade them to do what is right. (The Story of Alice Cullis; Or, How to Win by Example By Ellen L. Brown · 1881)