Haben die Sowjets die aserbaidschanische Grammatik manipuliert?

No. The Turkish elements from the pre Soviet era you are referring to have nothing to do with Azerbaijani having a different grammar in the past. They were never the standard language. For some reason, people like Rəsulzadə and Hüseyn Cavid decided to „Turkishify“ some of their writings. You can see this pattern a lot more in Cavid’s writings – take a look at Iblis, for example.

The Azerbaijani grammar was still the same. We have never used the „yor“ suffix in the past; actually, it didn’t even exist in the first place and evolved to „yor“ from something like „yürü“ or „dur/turur“ or something like that over time in Modern Turkish

Things like „istiyoruq“, „kəndi“, „qonuşmaq“ are simply copy pasta from Turkish of that era to likely gain readers from Turkey as well, in addition to local Azerbaijanis.

While Azerbaijani and Turkish grammars overlap 80-85 percent of the time, they still have their distinctions and yor is just one of them.

Who said we don’t use „anlamaq“? We actually do. Also, I am curious: don’t you guys say „düşünmək“ for „to understand“ in the South like Turkmens, Kazakhs (tüsin), etc.?

During the 1900s there was a fraction within the intellegentsia that aside from moving away from the perso-arabic literary traditions, consciously sought to highlight their turkish/turkic identity, and accordingly, were mixing their mother dialect with the ones that were spoken by the majority in anadolu. This mostly had to do with the rise of pan-turkism in the early 20th century, emphasized by İsmail Gaspıralı’s famous slogan: *“Dilde, fikirde, işte birlik“*

Rasulzadeh and many other notable azerbaijanis were one of them. What they did was borrowing Ottoman Turkish vocabulary, morphology, and stylistic forms, replacing Azerbaijani endings or words with Ottoman ones etc.
Probably the most well-known example of that is the national anthem that uses „hazırız“ instead of „hazırıq“. It was written as a poem by Ahmad Javad, a prominent persona in this movement, where he would create by combining both turkish and azerbaijani words, he also wrote Çırpındı Qara Deniz. All of these choices were an intentional move at emphasizing a shared turkic identity, as well as creating a modern and a secular language that the majority of the turkic peoples would understand.

However it also needs to be addressed that not everyone supported this idea. As an example, the father of pretty much everything that had to do with art and music in Azerbaijan, Üzeyir Hacıbeyli, believed that a national culture and prosperity should grow from people’s living language, not be imported, whether from Persia or İstanbul. He didn’t join the artificially created form of the language that ordinary azerbaijanis couldn’t really understand, he was advocating for a modernized version of the native azerbaijani instead, using words from all of the different languages, such as russian, french, arabic, ottoman turkish, but keeping the base and fundamentals purely azerbaijani.

What united all of these intellectuals was a shared love for their nation and a desire to move away from the Perso-Arabic cultural sphere and to modernize society, the difference lay in just how each of them chose to do so.