How to extract webpage data with a node and class in rvest

It looks like the information you are looking for is store in the "index.php" file that gets downloaded when the web page loads. The current link for Maine is "https://gasprices.aaa.com/index.php?premiumhtml5map_js_data=true&map_id=21&r=89346&ver=5.9.3".
I am not sure what the r=89346 value is for, maybe a timestamp, tracking id, temporary token (to prevent web scraping) etc. I suspect this URL will change thus you may need to use the developer tools on the browser to obtain the current url.
Also, map_id refers to state but I don't know the rational, Florida is 1, NC is 35 and Maine is 21.

Download this file, then extract the JSON data and convert. The data starts with a {"st1": and ends with }}.

library(dplyr)

#read the index_php file and turn it into character string
index_php <-readLines("https://gasprices.aaa.com/index.php?premiumhtml5map_js_data=true&map_id=21&r=19770&ver=5.9.3")
index_php <- paste(index_php, collapse = " ")

#extract out the correct JSON data part and convert
jsondata <- stringr::str_extract(index_php, "\\{\"st1\":.+?\\}\\}") 
data<-jsonlite::fromJSON(jsondata)

#create a data frame with the results
answer <- bind_rows(data)

      id name         shortname link  comment image color_map color_map_over
   <int> <chr>        <chr>     <chr> <chr>   <chr> <chr>     <chr>         
 1     1 Androscoggin ""        ""    $4.964  ""    #ca3338   #ca3338       
 2     2 Aroostook    ""        ""    $4.928  ""    #dd7a7a   #dd7a7a       
 3     3 Cumberland   ""        ""    $4.944  ""    #ca3338   #ca3338       
 4     4 Franklin     ""        ""    $4.936  ""    #dd7a7a   #dd7a7a       
 5     5 Hancock      ""        ""    $4.900  ""    #01b5da   #01b5da       
 6     6 Kennebec     ""        ""    $4.955  ""    #ca3338   #ca3338 

There are some extra columns which need removal, I leave it as an exercise for the reader.

So, you can gather the state info, including state level prices from the initial US page. You can also, from there, gather the urls for each state page. Make a request to each of those pages, and store the returned html. You can then, depending on whether the county data is in a php file, either extract the php file links, request that file and process out the info you want, or, in the case of no php file, extract the necessary data from the html already stored from the state requests.

Below extracts all the prices for all states and counties. There is a state DataFrame and a state with counties DataFrame.

library(tidyverse)
library(rvest)

get_data <- function(state, url) {
  # extract county and price data from php files. Pass in state abbreviation and php file URI. 
  s <- read_html(url) %>%
    html_text() %>%
    str_match("map_data\\s+:\\s+(.*\\}),") %>%
    .[, 2]
  return(
    tibble(
      state = state,
      county = s %>% str_match_all(',"name":"(.*?)"') %>% .[[1]] %>% .[, 2],
      price = s %>% str_match_all(',"comment":"(.*?)"') %>% .[[1]] %>% .[, 2]
    )
  )
}

start_url <- "https://gasprices.aaa.com/?state=US"
page <- read_html(start_url)

# get state price info and urls for state pages
data_strings <- page %>%
  html_text() %>%
  stringr::str_match('placestxt = (".*")') %>%
  .[, 2] %>%
  str_replace_all('\\"', "") %>%
  str_split(";")

df_state <- data.frame(subset(data_strings[[1]], lapply(data_strings, function(x) {
  x != ""
})[[1]]) %>% map(., ~ str_split(.x, ",")) %>% unlist(recursive = F)) %>%
  transpose() %>%
  .[c(1:4)] %>%
  set_names("abbr", "state", "price", "url")

state_data <- lapply(df_state$url, read_html)

# find the php file links
df_state$data_url <- lapply(state_data, function(item) {
  item %>%
    html_element("[src*=js_data]") %>%
    html_attr("src")
})

# separate out dataframe according to whether county data is in php file or in previously stored html
no_valid_data_url <- df_state %>% filter(is.na(data_url))
has_valid_data_url <- df_state %>% filter(!is.na(data_url))

# grab the data for states where there are php files with county info
df_state_county <- map2_dfr(has_valid_data_url$state, has_valid_data_url$data_url, get_data)

# add in missing info i.e. #  handle cases where data_url is NA e.g. https://gasprices.aaa.com/?state=DC
if (nrow(no_valid_data_url) > 0) {
  html_to_use <- state_data[match(no_valid_data_url$abbr, df_state$abbr)]
  df_state_county_no_data_url <- map_dfr(html_to_use, function(html) {
    state_node <- html %>% html_element(".selected")
    state_text <- state_node %>% html_text(trim = T)
    return(
      data.frame(
        state = state_text,
        county = state_text,
        price = html %>% html_element('td:contains("Current Avg.") + td') %>% html_text()
      )
    )
  })
  df_state_county <- rbind(df_state_county, df_state_county_no_data_url)
}


head(df_state, 2)
head(df_state_county, 2)