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"The lyrics, which were all written by Leach" - think you need to put the full name/position...
"Without wasting time" seems too informal.
Think you only need to put "Roadrunner Records" the first time, you can go by "Roadrunner" the other ones.
"Bassist Mike D'Antonio created artwork for the label, and asked as a favor for Killswitch Engage to be signed, subsequently the label accepted. Ferret Records CEO, Carl Severson signed the band without a contract" - prose could be better. Maybe "The band got its recording deal as a favor for the artwork bassist Mike D'Antonio created for Ferret. The label's CEO, Carl Severson, signed Killswitch Engage without..."
The end of the first paragraph lacks the ref.
"Gitter offered to sign the band to the label, but the band was apprehensive at first, judging by the current artist roster of Roadrunner Records. Gitter felt that signing Killswitch Engage was the label's last chance to have any heavy metal bands on the roster. After nearly six months of deliberation, the band signed to Roadrunner Records due to the opportunities provided for them and began work on their major label debut, which would become Alive or Just Breathing." - Think you could merge Gitter's offer in the first sentence and his reason in the second one into a single sentence. The band's reluctance (by the way, "current" might not fit with past events - maybe "Roadrunner's artist roster at the time"?) and the negotations can serve as another one. Then a third sentence regarding that work begun.
For the first paragraph how about "recorded a demo produced by drummer/guitarist Adam Dutkiewicz, which featured the songs..."
The second paragraph could have the first two sentences merged. Also, "to be with his family" -> "to spend more time with his family".
You put "completing the album" and "completion of the record". You'd better rewrite to show the first was finishing composition. The second might be moved to Recording as it regards the recording sessions themselves, and the future drummer is mentioned there.
Wikilinks on guitar riffs, moshy, harcore, and thrash metal would help.
"there was too much negativity in the lyrics of music in general" -> "music lyrics in general featured too much negativity"
Perhaps the last paragraph can be moved into "Recording"?
"During the Alive or Just Breathing recording sessions, Killswitch Engage recorded 15 songs. Three of the songs were re-recordings of songs from the band's debut album. Stroetzel stated that the band decided to re-record "Temple From the Within", "Vide Infra", and "In the Unblind" because at the time, Killswitch Engage was unsure of how many people would ever hear Killswitch Engage because it was released through a small label." - you can list the titles after "Three of the songs", and rewrite the following sentence (specially to include what is listed in the ref about "tak[ing] them a little further this time").
"to stay in tune better" -> "improve the tuning", perhaps? The following sentence might be cut.
"Jesse's home" ->"Leach's home"
"Mike Gitter was unhappy about this when he heard of it as Killswitch Engage turned the record into Roadrunner" -> can you rewrite for something along the line of "Mike Gitter was reluctant to hear about this when receiving the record from Killswitch Engage"
"Dutkiewicz's sister, Becka" -> put the comma after her name
"Killswitch Engage, as well as Roadrunner Records was very pleased with the mix. Dutkiewicz specifically..." -> The band and the label were very pleased with the mix, and Dutkiewicz in particular..."
Artwork and packaging
Merge the first two sentences. (e.g. "Mike D'Antonio owns a graphic design company called Darkicon Designs, and has created artwork for bands such as Shadows Fall, All That Remains and Crowbar.")
"As Killswitch Engage was signing to Roadrunner Records, the band made sure that written into the contract was for D'Antonio to produce all of Killswitch Engage's" -> "Killswitch Engage made sure to have written in their contract with Roadrunner that D'Antonio would produce all of the band's artwork."
The last sentence, regarding D'Antonio doing the rest of the band's artwork, could be cut.