The classical Sankoff algorithm for the simultaneous folding and alignment of homologous RNA sequences is highly influential, but it suffers from two major limitations in efficiency and modeling power. First, it takes $O(n^6)$ for two sequences where n is the average sequence length. Most implementations and variations reduce the runtime to $O(n^3)$ by restricting the alignment search space, but this is still too slow for long sequences such as full-length viral genomes. On the other hand, the Sankoff algorithm and all its existing implementations use a rather simplistic alignment model, which can result in poor alignment accuracy. To address these problems, we propose LinearSankoff, which seamlessly integrates the original Sankoff algorithm with a powerful Hidden Markov Model-based alignment module. This extension substantially improves alignment quality, which in turn benefits secondary structure prediction quality, confirmed over a diverse set of RNA families. LinearSankoff also applies beam search heuristics and the A$^\star$-like algorithm to achieve that runtime scales linearly with sequence length. LinearSankoff is the first linear-time algorithm for simultaneous folding and alignment, and the first such algorithm to scale to coronavirus genomes (n $\approx$ 30,000nt). It only takes 10 minutes for a pair of SARS-CoV-2 and SARS-related genomes, and outperforms previous work at identifying crucial conserved structures between the two genomes.