reCAPTCHA timing out on verify

Question

Having made no changes (but also not tested this in a couple months), my reCAPTCHA code is timing out when trying to verify the user response (both on localhost and on our DEV server). The IP is rotating in the error message (216.58.218.228:443, 172.217.1.4:443, others). I would normally think that this is a firewall issue - I can't telnet to the IP/Port combinations in the error messages - but I can pull up https://www.google.com/recaptcha/api/siteverify in a browser and get an error response back quickly. Any ideas on what might be going on?

Code I normally use:

// Prepare input.
var input = new NameValueCollection
{
    { "response", captcha },
    { "secret", _settings.SecretKey }
};

// Make call.
byte[] response;
using (WebClient client = new WebClient())
{
    response = client.UploadValues(_settings.VerificationUrl, Http.Verbs.POST, input);
}

// Parse response.
var body = Encoding.UTF8.GetString(response);
var result = JsonConvert.DeserializeObject<VerificationResponse>(body);
return result.Success;

Attempt at verifying through GET (as recommended here):

using (WebClient client = new WebClient())
{
    var response = client.DownloadString($"https://www.google.com/recaptcha/api/siteverify?secret={_settings.SecretKey}&response={captcha}");
}

Example full error message:

A connection attempt failed because the connected party did not properly respond after a period of time, or established connection failed because connected host has failed to respond 216.58.218.228:443