Initially, this may sound like bad advice, but when you think about it, an animal who is gone (or a person, for that matter) would not want us to put our lives on hold for them (as they are not coming back). SO, while mourning is healthy, sometimes moving on is easier when you have another animal/person in your life. It isn't a replacement, it is an opportunity to make another life better and help a creature in need. I think that in death, everything "human" (or "worldly") probably seems a little bit more silly than it does to the living...I mean, after a death, the living often guilt themselves out over the mere concept of feeling emotions for other creatures, as though it is some sort of adulterous affair (when really, there is no cap on the love we can feel for others----it's not like you can only love one animal/person etc---dead or alive).
When our dog died years back (from cancer), I was grief stricken...I had spent days sleeping on a tile floor next to her and it was really bad. The day after she passed, I went to the pound and everyone in my family was SERIOUSLY offended, but here is the thing: I will never be over that loss. Time will make it easier, but it doesn't ever go away totally, so waiting seemed fruitless (because it still hurts when I think about it). We all planned to get a new dog eventually anyway, and dogs are killed in pounds daily (because no one comes). I didn't see the point in waiting, as it wouldn't bring her back and I needed something to do with all of my grief/anxious/sad energy.
People like to wait (or refuse to get another animal) out of reverence, but reverence is a social construct of the living to some extent. Despite my family's objections, we ended up picking a dog that day at the pound (and NO ONE thought that would happen, as I was the only one who went originally----had to call everyone 50 x to get them to come meet me). The dog we got was AMAZING and she would have been euthanized the following day had we waited...It didn't replace are negate the loss, but it did provide a channel for emotions aside from wallowing etc.
Sometimes, you have to put the injured part of your heart to good use in order for it to stop hurting so much. It doesn't mean that you don't love the one you lost---and any human or animal with a consciousness after death would absolutely have no problem with you continuing to live your life/providing a home for another animal. Taking time to mourn is healthy, but at the same time, the longer you do it, the harder it can be to move on without feeling like you are dismissing the power of your memories with that person/animal (even though that isn't the case).
Losing a pet can be worse than losing a human family member in some cases (because even the people we love can do mean things on purpose) but animals do not necessarily work that way. I get what you are saying 100%..It is really hard.
You don't have to go into getting another animal (or entering another relationship in general) as a means of replacing or covering up others--that is not the point (again, there is no maximum on the love we have to give to others). My point is to find joy in putting your emotions into something living...Not a replacement, but a new opportunity to do good and help those in need---you may never feel ready, but I'm not sure many people ever really do (until the try).
Bottom line: Every worthwhile relationship IS going to hurt at some point (because everything dies, ends, changes etc). There is no way to make a loss any less painful, because that pain means there was love (and that is worth something). That having been said, DO NOT LET THE PAIN STOP YOU FROM LOVING/LIVING etc. I am sorry you are going through all of this