Definitely shop around. Rates and general terms vary. Ours has a variable rate which is something like 0.6% right now. But we can lock in some or all of any balance (up to 5 different locks at any given time) whenever we want. Someone else might offer us more credit, or a lower rate on the locked portion, or something else, but we liked these terms. Definitely worth shopping around and asking about different products even from the same company.
If you want to "bump up the limit" you will likely have to reapply, and that could mean losing the favorable terms you have. We wanted to increase ours when we were paying off our traditional mortgage. (We hoped to be able to pay off the mortgage with the HELOC at .6%.) However, that product is no longer available to in order to increase the limit, we'd have lost the good deal. We decided to keep what we had and find another way to pay off the rest of the mortgage. Our limit is nearly $100k, so it is plenty for just about anything, but still far, far below the amount of equity we have.
The larger point is that it may not be necessary to max out the HELOC. For us, nearly 6 figures is more than enough. IF we need more money than that, the economic shit has really hit the fan and our property value likely will have tanked, meaning the terms would probably be adjusted down anyway. So if you have a decent amount of equity now (or when you apply), don't worry about maximizing every penny.