How did the Elders and the people they met keep Easter?
If there is evidence of an Easter observance by anyone, missionary or otherwise, in the six diaries, I haven’t found it.  It is possible that the meeting, singing, preaching, and visiting the Elders did and observed on Easter Sundays were prompted or amplified by Easter, but I can’t tell it from the text. 
Only one entry reaches beyond “consistent with” to “possibly suggestive of”: on Easter Sunday 1901 (Apr 07), Elder Brooks preached the funeral of church-member William Williamson, who had died the previous November. To delay a funeral 4.5 months and then “just happen” to land on Resurrection Sunday suggests the possibility of intent. 
A low-key Easter might have been part of a larger trend. At General Conference on Easter Sunday 1904 President Duffin was impressed by Church President Joseph F. Smith’s talk. Although the talk strongly affirmed Christ’s divinity, it was not an “Easter sermon” and only two speakers mentioned the holiday by name. 
To be clear, though the Elders do not mention Easter in the diaries, they do mention Christ and the resurrection. They identified themselves as “ministers of the gospel of Jesus Christ,” heard instructions on “bearing testimony to the world of divinity of Jesus Christ,” and preached “on the life of Jesus.” They criticized and/or argued with those who understood Christ differently and those “who professed to be a follower of Jesus Christ” but did not provide food or lodging. They prayed in the name of Christ and, as Elder Clark wrote, “I am glad that I am worthy to be persecuted for a testimony of Jesus Christ and I pray god that he will give me Strength to hold out faithful to the end.”
The “Southwestern States Mission” series uses the diaries of six missionaries who served in eastern Texas around 1900 to illustrate aspects of Mormon material culture, lived religion, and social History. The missionaries are Mission President Duffin and Elders Brooks, Clark, Folkman, Forsha, and Jones. The series is inspired by Ardis Parshall’s serial posting of the missionary diary of Willard Larson Jones at Keepapitchinin. Previous installment here.
 I looked for patterns or words related to Mardi Gras, Ash Wednesday, Lent, Good Friday, Easter, Gethsemane, Golgotha, Calvary, Crucifixion, Resurrection, and so on. With the eventual migrations of large numbers of Cajun, Hispanic, and South Vietnamese Catholics to Southeast Texas, later missionaries probably had a harder time not noticing the season between Mardi Gras and Easter.
 Most of the entries are vague enough to be consistent with Easter observance, but Elder Clark’s 1900 meeting on Easter Sunday had a non-Easter hymn and topic.
 They might have been waiting for an Elder to provide a “Mormon” service or for Elder Brooks in particular. The funeral was three days after Brooks returned to the Williamson’s area. It is possible (but I think unlikely) that no missionaries had visited since William died. Even if they had, the Williamsons and Elder Brooks were close (at least as reported in Elder Brooks’ diary).
 President Smith’s sermon is in Conference Report, 1904 April, p 1-5. I did not read any of the talks in full; I am relying on non-corrected OCR and text search so might have missed some. The talk after Smith’s was an “Easter sermon” by Anton H Lund, Second Counselor in the First Presidency (p 5): “I am reminded that this is Easter. We celebrate today the glorious event, of the resurrection of our Savior. About 1870 years ago this great event, which confirms our hope of eternal life, took place.”
The following day, SB Young responded: “Reference was made yesterday to the life of our Savior, and to the generally accepted idea that yesterday was the anniversary of the day on which He rose from the tomb. However, whether Easter Sunday is the proper day or not, we have no objection to it. The great underlying fact is that He rose from the dead, after having suffered crucifixion for the sins of the world, and that He was the Son of God. … When Easter Sunday comes, or when any other Sunday comes, it is the Lord’s day and we reverence His high and holy name. We come together and worship Him. Thus to us every Sunday, in a sense, is Easter Sunday, for we worship the Lord on that day, and not only on Sunday, but every day in the week.” (Seymour B Young [Seven Presidents of the Seventy, First President], Conference Report, 1904 April, p 37.