Old English does not sound exactly like Dutch. There are periods of influence however the best way you can explain it is
1) ancient Germanic languages were a dialect continuum from the Alps to lower scandinavia. Eventually island scandinavians such as the Gotlanders mass migrated to the Ukraine and became the Goths and East Germanic is the first to significantly diverge. East Germanic went extinct in the 1700s when the Crimean Goths were absorbed into the Crimean Tatars.
2) Angle and Saxon and Jut (who gave their name to Jutland, the mainland part of Denmark where no one lives) were from the flat, low part of modern Germany/Denmark/Netherlands (Frisians originally occupied most of the Netherlands.
3) Old Dutch is the stage immediately after High Frankish, the language Charlemagne would have spoken to his dukes alongside Medieval Latin and Old French. Frankish was a 'middle Germanic' language and from the region immediately south of the original Saxon homeland- modern Franconia has many related dialects.
4) modern German is derived from the prussianized form of High German, the court dialect of Habsburg Wien/Vienna when it was the greatest city in Europe (this is why Bavarian is still extremely 'distinct' from national German), called High German because it comes from the Alpine regions.
5) English also has many linguistic similarities with Scandinavian languages both because it was the 'west germanic' dialect bordering 'north germanic' dialects like proto-Danish as well as being invaded by Danes and Norwegians multiple times.
6) England was also invaded by the Netherlands as well as trading extensively with the netherlands during the time of the Hanseatic league.
7) it is thought the rise of High German over the Saxon of the Hanseatic league is because the League collapsed, otherwise 'German' would be more similar to Dutch and English.
8) Dutch has been High Germanized over time due to proximity while English is conservative.