Old English had full-stop punctuation in some manuscripts, but did not have regular sentence-beginning capitalization.
According to Introduction to Manuscript Studies, by Raymond Clemens & Timothy Graham (Cornell UP, 2007), the punctus, the forerunner of modern periods and commas, would have been present in manuscripts in Latin during the Old English period (p. 83). Early manuscript punctuation used the system of distinctiones, which were used to guide oral recitation:
distinctio - a punctus at the height of the top of the letter. Ends a sentence or section.
subdistinctio - a punctus at the bottom of the line. Signals a minor break.
media distinctio - a punctus at mid-line. Signals an intermediate break, like for a sentence or clause.
The distinctio and media distinctio were full-stops for sections or sentences, though how consistently it was used varied by scribe and manuscript. Clemens and Graham's explanation of usage includes a nod to Old English scribes, who would have sometimes used these marks:
To further clarify meaning, both Irish and Anglo-Saxons punctuated assiduously, using the distinctiones system. A notable (and quite late) use of the system in its basic form, with a single punctus placed at differing heights, occurs in the well-known [Latin MS] Bury Psalter (Vatican City, Biblioteca Apostolica Vaticana, MS Reg. lat. 12), written in the early eleventh century, probably at Canterbury. There the high point is used at the end of each verse of the Psalms, the midpoint is used to mark the break at the middle of each verse, and the low point is used to mark off minor sense-units within each half of the verse. (p. 84)
So scribes in England and Ireland were writing Latin manuscripts with distinctiones that functioned and looked like elevated commas and periods for most of the Old English period. It turns out that these marks do appear in Old English manuscripts:
in the Exeter Book from the second half of the 10th century, as featured on the British Library website, both media distinctio (mid-line dots) and distinctio (two dots and a long mark - looks like :7) are visible. Capital letters mark sections of text.
in Cotton Vitellius A.XV, f.95r (British Library), a distinctio is visible at the end of line 7 (.,), and a single media distinctio (.) is visible after the first word in line 5. Capital letters mark sections of text.
in Vercelli f. 104v (Digital Vercelli Book), the distinctio (:7) visible at end of section, with media distinctio at the end of the second line after hwaet and periodically thereafter. Capital letters mark sections of text.
Note that the only capital letters present are used to denote the beginnings of sections of text, not sentences, suggesting that period-like punctuation appeared long before regular sentence-starting capital letters. Let's look back at Vercelli and the first several lines of transcription:
7 HWæt ic ſwefna cẏꞅt secgan wylle hæt mege mætte
8 to midre nihte syðþan reord berend reste wunedon.
9 þuhte me þæt ic ge ꞅawe ꞅyllicre treow onlyft
10 lædan leohte be wunden beama beorhtoꞅt eall þæt
11 beacen wæꞅ be goten mid golde gimmaꞅ ſtodon fægere
12 æt foldan ſceatum . ſwylce þær fife wæron uppe
A period at the end of line 8 and another period at the middle of line 12 marks two sentences, and these same marks are visible in the manuscript. The second sentence does not start with a capital letter. Nor does the third. So while punctuation systems remain flexible for a few centuries afterward, and they were applied to many units of meaning besides sentences, Old English scribes already used punctuation to distinguish sentence level units.
The regular capitalization of initial words in a sentence takes longer to form, and isn't present in Old English.