There are plenty of alternatives, but they’re also shared hosts, so they’ll have similar problems as well.
I can give an example from my own experience. Site5 is another well-known host. Not sure how they compare in size to DH, but I’d guess that most people that heard of one, also heard of the other.
I have an account there, in addition to my DH account. I’m happy with them, as I am with DH, but I’ve had email problems, brief site outages, and DNS issues there. Plus, I pay more there, get less space/bandwidth, and fewer features.
Would I recommend them? Sure - I even have this nifty referral link!. And here’s my DH referral link for when you come back.
I’m not with them because I don’t like DH, or vice versa–I just don’t like having all eggs in one basket.
If you’re set on leaving, I’d limit your choices to companies that have been around awhile, then go from there as far as features and pricing are concerned.
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