Dear Alexandra Sokol:
I have tried to search for a "D. Folis" and for a "D. Tolis" in the interface for the Ellis Island database at heritage.statueofliberty.org (one needs to register to see ship manifests, but registration is free) , and I have not had luck finding your "David Folis/Tolis" there: the only Folis-es or Tolis-es with a first initial D that I found were Greek or Hungarian. (Therefore, probably not Jews from the Russian Empire.)
I had somewhat-better luck looking for a David Sokol (while the interface found me a David Sokol who arrived in 1904 on the ship König Albert, the manifest image it gave for him didn't have him). I used Ancestry.com to try to find an immigrant David Sokol, and succeeded in finding this David Sokol -- an 18-year-old (if I read correctly, his occupation was given as "tanner") from what may have been written "Krinick" (?) in the Russian Empire (which may have been the town (said to be 28 miles east of Bialystok) whose name is officially spelled "Krynki"). (It seems that he stated he was coming to join an uncle whose last name was written "Salamon" (I can't decipher the 1st name as well) in Newark, New Jersey.)
(My maternal grandmother's parents ("Paat" on immigration records in 1898; later known as "Pat", "Patt", and (eventually) "Pate" in the US) emigrated from Bialystok, but I have learned within the past year or 2 that my grandmother's father's Pat family may have come from Krynki, so I've learned a tiny bit about it -- and apparently it was known for leather-production and tanning.
I'm not sure in which era there was a "Polish/Russian border" -- although Krynki is now near the Polish/Belarusian border, it was (for years before World War I) definitely within the Russian Empire, and I'm not sure what kind of "border" which smugglers would get over would have been near it before World War I. (Maybe the recollection is from 1918 or later?).)
I didn't find any other David Sokols on the Ellis Island database who arrived before 1917, but Ancestry.com has a 3-year-old "Dudio" Sokol who arrived in 1905 (seemingly with the mother), a (9-year-old) David "Sokel" (I think) who arrived in 1906, an 18-year-old David "Sokal" who arrived in 1910 from "Lemberg" (today's Lviv) in Galicia (not in the Russian Empire before World War I), and a 21-year-old "tailor" named David Sokoll who arrived in 1910 said to be from "Psholenka" in the Russian Empire, who stated that he was coming to meet his brother-in-law Adam Stribel (I think) who lived on Canal Street in lower Manhattan.
I think that the David Sokol who came from "Krinick" (and possibly Krynki) is more likely than anyone else I found today; I hope this research will be of help to you (and maybe others who will read this).
(I don't think that I have found much luck finding your great-grandmother "Fagabluma Sokol"; I have found a (34-year-old) Feige Gittel Sokol who arrived in 1911 (stated to have emigrated from Bialystok) with a young son and daughter -- who was said to be meeting a husband in Brooklyn.)
Best wishes for the fall and/or for Jewish Year 5781.
New York, NY
(researching Paat/Pat/Patt/Pate from Bialystok (and possibly Krynki), Poland, Kornhauser from Turka (now in Ukraine) -- and possibly Stefkowa (now in Poland), Kantor (probably from Bratslav -- now in Ukraine), Gelperin/Halperin (emigrated from today's Vilnius, Lithuania; father probably from a Krasnoye now in Belarus) -- and (to some degree) related families (including Jaffe in today's Lithuania -- ancestral to Mrs. Gelperin/Halperin (my great-grandmother) )
PS: As you (Alexandra Sokol) may soon find (if someone else hasn't posted about this before me) very few names were "changed in Ellis Island" -- as American immigration workers were instructed to not change the names given on passenger manifests.