The surname "Dapsis" is probably spelled "Daps^ys" and pronounced dahp-SHEES. The emphasis could easily shift back among non-Lithuanians because so few English words are accented on the final syllable. The "s^" with a little birdie over it is pronounced "sh" as in the English word "sheep." It is typed "s^" on non-Lithuanian keyboards. It is unlikely that it was shortened from "Donavich" or anything beginning "don-", which indicates a different name entirely, perhaps someone's maiden name.
In the current online phone book for Lithuania http://www.zebra.lt/lt/suzinok/telefonai) which is unfortunately only in Lithuanian, there are these listings for the surname Daps^ys (there are only a few listings for Daps^as):
Alfonsas Daps^ys
J. Janonio g. 12-11
Tels^iu miesto sen.
Tels^iu m.
844432286
Antanas Daps^ys
Z^irmunu g. 40-30
Vilniaus m.
852797265
Bronius Daps^ys
Mindaugo g. 13-1
Anyks^iu sen.
Anyks^c^iu m.
838151916
Dalius Daps^ys
P. Mas^ioto g. 43-6
Pakruojo sen.
Pakruojo m.
842151566
Danielius Daps^ys
Gelez^inkelio g. 1-10
Vilniaus m.
852611640
Gediminas Daps^ys
Statybininku g. 11 - 15
Panevez^io m.
845440553
Henrikas Daps^ys
Rudes g. 8-86
S^iauliu m.
841435482
Jonas Daps^ys
Aus^ros g. 3
Kretingos miesto sen.
Kretingos m.
844554513
Juozas Daps^ys
Kairis^kiu k.
Papiles sen.
Naujosios Akmenes raj.
842550052
Jurgis Daps^ys
J. Matulaic^io a. 5-3
Vilniaus m.
852301793
Kastytis Daps^ys
Parko g. 61-1
Vilniaus m.
852679713
Kstutis Daps^ys
S^iaures pr. 3-15
Kauno m.
837712211
Pranas Daps^ys
Sodo g. 49c
S^iauliu m.
841544485
Vaclovas Daps^ys
Klaipedos g. 10
Tels^iu miesto sen.
Tels^iu m.
844474964
Plus the following for unmarried women whose father was named Daps^ys:
Apolonija Daps^yte
Ramygalos g. 82-33
Panevez^io m.
845432945
Birute Daps^yte
Z^aliuju Ez^eru g. 11-54
Vilniaus m.
852701439
Danute Daps^yte
Ramuc^iu g. 39-16
Naujosios Akmenes miesto sen.
Naujosios Akmenes m.
842556095
Danute Daps^yte
Dainu g. 49-20
S^iauliu m.
841417450
Elena Daps^yte
L. Vaineikio g. 23
Palangos m.
846053368
Regina Daps^yte
Rygos g. 25-14
Vilniaus m.
852443264
Plus another 37 listings for women married to a man named Daps^ys or Daps^sas (the ending for married women's names is the same regardless of the ending of her husband's name).
The abbreviations are:
g. = gatve, street
m. = miestas, town or city
sen. = seniunija, elderate or eldership (a kind of administrative division larger than a town, but smaller than a city or district, perhaps like an American township or borough or an area of a city)
miesto sen. = miesto seniunija, elderate of the town
k. = kaimas, village
raj. = rajonas, district (compare to the English "region")
The endings "-o", "-u", "-aus", "-os" all indicate the possessive case, meaning "of or belonging to" as in the phrase "city of Vilnius" = "Vilniaus miestas".
This is a fairly long list of possible relatives, but it is a good bet that some of them, at least, are related, however distantly, to your family. Once you find out where the Daps^ys family was from, you might be able to narrow the above list down significantly.
The given name Casimir in Lithuanian is Kazimieras, but in America it was very commonly changed to Charles -- for reasons no one can explain exactly, just as the Lithuanian Vincentas often became William in America, even though it means Vincent.