Alternatively
On the other hand
Albeit
Since
Inasmuch
In contrast
Contrarily
Conversely

You can still use your preferred conjunctions; just use them to join the sentence rather than ending the sentence and beginning a new one.
For example:
It may seem as though the activity is futile, but scientists believe there is an overlooked purpose behind it.
Also, try using semi-colons.

This is just Grammarians being old f*rts!

Like never ending a word with a preposition!

To which Churchill (no mean wordsmith in his own right) once responded "That is something up with which I will not put".

Or words to that effect.

Rules applied retrospectively (which is the grammarians' whole tawdry stock in trade) are just that - already outmoded and phoney.

Living language and real language is what people use - and always has been - so that makes it right and makes the Grammar Gestapo wrong.

Use language to communicate and let the grammar obsessives go look up the word 'communication'.

It matters not one bit that you are formally correct or incorrect - in ten years' time grammarians will come round to suddenly noticing that we have all been starting sentences with 'But' etc for centuries.

Who gives a hoot unless your pay packet only drops into your hand because you are a dried-up grammar teacher?

Did Shakespeare swot up a grammar book?

Does a five year old child who can speak perfectly good English curl up in their cot with a grammar primer? No, I thought not.

Ban all grammarians! Off with their participles!