You can use "e'er" anywhere you would normally use ever, but it is a term used very often. Even back in the 1800s, when it was used more often, the word "ever" was very much more popular. It is rarely found outside of hymns or poems, and usually ones written long ago.
If you are writing a poem or song and need the word ever to be one syllable rather than two, you could use it then, or if you are specifically looking for an archaic voice you could try it, but in any normal circumstances, you should use "ever" instead.
Here are some examples of e'er in a sentence from Google Books:
1807: "If e'er my muse should incense such a name, / Then may my steps, forsake the paths of fame."
1808: "When e'er I view my little boy / Such innocence I find.
1848: "I press'd her fondly to my breast, / And swore it o'er and o'er, / That she this night with me should rest, / Nor e'er meet sorrow more!"
1913: "May nought ever grieve thee, / May no one deceive thee / Where e'er thou shalt go."
There are some examples from later, but all the ones that I can find are just quoting poems, plays, or songs written earlier.
Some words that end with -eer include engineer, volunteer, and pioneer.
Words that have the letters -eer in them are deer, beer, and cheer. Other words that have the letters -eer in them are auctioneer and career.
The suffix -eer changes the noun 'election' to the verb electioneer.
Reindeer
eer
engineer
Used in texts, "eer" means ever erring which implies ever making mistakes.
Extended ER diagram is a variant of ER diagram. EER include ER diagram + hierarichal concept. Extended ER diagram is also called Enhance ER diagram(EER).
LG LWHD1500ER 15000 BTU Window Air Conditioner has an energy efficiency rating of 12.0 EER, compared with other products that mostly do not pass 10.0 EER (EER meaning "Energy Efficiency Ratio" and a higher EER means that the air conditioner is more efficient)
estimated energy requirements
eer
auctioneerbeercareercheerdeerengineerfreerjeeroverseerpeerpioneerpuppeteersneersteerseersheervolunteer